RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php. Selfie Fractions: Addable and Dottable To
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
Selfie Fractions: Addable and Dottable Together Received 10/07/16
Inder J. Taneja1 Abstract
A addable fraction is a proper fraction where addition signs can be inserted into numerator and denominator, and the resulting fraction is equal to the original. The same is true for dottable fractions, i.e., instead of additions we have multiplication. In this work we have written fractions having both the operations, i.e., addition and multiplication. The work is for different digits, i.e., there is no repetition of digits in the same fraction. Also, the numerator is less than denominator.
1
Introduction
Kieth [3, 4] for the first gave an idea of dottable fraction. It is a proper fraction where multiplication signs can be inserted into numerator and denominator, and the resulting fraction is equal to the original. Keith [3, 4] idea was only with multiplication. For the first time, we extended it to other operations also, such as with addition, multiplication, potentiation, etc. We can think all of them together also. See below some examples in each case: • Addable Fractions
• Subtractable Fractions
• Dottable Fraction
• Dottable with Potentiation Fractions
9+6 182 18 + 2 96 = , = , etc. 352 3 + 52 6734 6 + 734
(1)
20 − 4 726 72 − 6 204 = , = , etc. 357 35 − 7 1089 108 − 9
(2)
1 × 3 416 4 × 16 13 = , = , etc. 624 6 × 24 728 7×2×8
(3)
95 9×5 728 72 × 8 = 4 , = , etc. 342 3 × 2 1456 14 × 56
(4)
4980 4 − 9 + 80 3249 (3 + 24 ) × 9 = , = , etc. 5312 5 × (3 + 1)2 5168 (5 − 1) × 68
(5)
• Mixed Fractions: All Operations
Observing the examples given in (1)-(5), the numerator and denominator follows the same order of digits in both sides of each fraction. These kind of fractions, we call selfie fractions. There are two types of situations. One when all digits appearing in each fraction are distinct and second, when there are repetitions of digits. Initially, we shall work with distinct digits. Due to big number of fractions, later we shall work with repetitions. The work on addable, subtractable, dottable and dottable with potentiation can be seen in Taneja [28, 29, 30] and is separated as: 1. 2. 3. 4. 5.
Selfie Fractions: Addable - Equation (1) - [28]; Selfie Fractions: Dottable and Pontentiable - [29]; Selfie Fractions: Addable and Dottable Together - this work ; Equivalent Selfie Fractions: Dottable, Addable and Subtractable - [30]; Equivalent Selfie Fractions: Addable and Dottable Together - [31]. In the mixed type selfie fractions, if we consider only with addition and multiplication, the results are of type: 612 6 × 12 387 3 + 87 = , = , etc. 748 (7 + 4) × 8 2064 20 × 6 × 4 1 Formerly,
Professor of Mathematics, Universidade Federal de Santa Catarina, 88.040-900 Florian´opolis, SC, Brazil. E-mail:
[email protected].
RGMIA Res. Rep. Coll. 19 (2016), Art. 115
(6)
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
Still, we have equivalent selfie fractions with the same operations, such as, 287 28 + 7 2+8+7 1069 (10 + 6) × 9 1 + 069 1 × 06 + 9 = = , = = = , etc. 1435 (1 + 4) × 35 (14 + 3) × 5 7483 7 × 48 × 3 7 + 483 7 × (4 + 8 + 3)
(7)
The difference between equivalent selfie fractions and symmetric equivalent selfie fractions is that in the first case there are more than one representations for each fraction, while in the second case, by changing one operation by another another, still the fraction remains the same. For more details see author’s work [29, ?]. In this paper, our aim is to work with selfie fractions with simultaneous use of addition and multiplication as given in expressions (6). The equivalent fractions similar to expression (7) are given in Taneja [?]. Each fraction is with different digits, and numerator is always less than denominator. The study on numbers in different situations, refer author’s work [7]-[27]. For other studies see also [1, 2, 5, 6].
2
Selfie Fractions: Simultaneous Use of Addition and Multiplication
This section deals with the selfie fractions working with the operations of addition and multiplication simultaneously. Since there are may fractions, we have divided in small subsection, according to position of zero in different categories. The division of subsections is as follows: 2.1 2.2 2.3 2.4 2.5
2.1
Without Zero and Extendable to Ending in Zero; Without Zero and Non Extendable to Multiple of Ten; Fractions With Zero in Between; Fractions Ending in Zero in Numerator; Fractions Ending in Zero in Denominator.
Without Zero and Extendable to Ending in Zero
There are many selfie fractions that can be extended just multiply by 10 either in numerator or in denominator or both. Still, they remains selfie fractions. For simplicity, let us consider following three examples: •
12 1+2 = . 384 3×8×4
•
2×7×3 273 = . 468 4 + 68
•
217 2×1×7 = . 9486 (94 + 8) × 6
Multiplying 10 in denominator of first one, numerator of second one and numerator and denominator both of third one, still we have selfie fractions given as follows: •
12 1+2 = . 3840 3 × 8 × 40
•
2 × 7 × 30 2730 = . 468 4 + 68
•
2170 2 × 1 × 70 = . 9486 (94 + 8) × 6
•
217 2×1×7 = . 94860 (94 + 8) × 60
This subsection brings those selfie fractions that can be extendable ending in zero. Since the extended fractions are obvious, they are not written in this work. Moreover, this process happens with all dottable fractions [?]. Below are these kind of selfie fractions:
•
12 1×2 = . 54 5+4
•
15 1×5 = . 27 2+7
•
17 1+7 = . 85 8×5
•
18 1+8 = . 36 3×6
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
52 5×2 = . 78 7+8
•
87 8+7 = . 145 (1 + 4) × 5
•
189 18 + 9 = . 735 7×3×5
•
357 (3 + 5) × 7 = . 816 8 × 16
•
12 1+2 = . 384 3×8×4
•
126 12 + 6 = . 385 (3 + 8) × 5
•
189 1+8+9 = . 756 (7 + 5) × 6
•
378 3+7+8 = . 924 (9 + 2) × 4
•
12 1×2 = . 594 5 + 94
126 12 + 6 • = . 735 7×3×5
•
192 (1 + 9) × 2 = . 864 86 + 4
•
423 (4 + 2) × 3 = . 517 5 + 17
•
1×5 15 = . 297 2 + 97
1×2+7 127 = . • 635 (6 + 3) × 5
•
196 1×9+6 = . 784 (7 + 8) × 4
•
426 4×2×6 = . 781 7 + 81
•
2+1 21 = . 385 (3 + 8) × 5
215 2×1×5 = . 387 3+8+7
•
128 1×2×8 = . 576 (5 + 7) × 6
•
(4 + 5) × 6 456 = . 912 9 × 12
(2 + 1) × 5 215 = . • 473 (4 + 7) × 3
•
458 (4 + 5) × 8 = . 916 9 × 16
•
459 45 + 9 = . 612 6 × 12
•
462 4 + 62 = . 735 7×3×5
•
465 (4 + 6) × 5 = . 837 83 + 7
•
(4 + 7) × 3 473 = . 516 (5 + 1) × 6
•
483 4+8×3 = . 759 7×5+9
•
532 (5 + 3) × 2 = . 798 7+9+8
•
534 5+3+4 = . 712 (7 + 1) × 2
•
5 + 61 561 = . 748 (7 + 4) × 8
312 3 × 12 = . 468 46 + 8
•
612 6 × 12 = . 748 (7 + 4) × 8
315 3 × 15 = . 462 4 + 62
•
6×5+1 651 = . 924 (9 + 2) × 4
315 3 × (1 × 5) = . 924 (9 + 2) × 4
•
14 1+4 = . 2856 2 × 85 × 6
21 2+1 • = . 735 7×3×5
•
•
2×1 21 = . • 756 (7 + 5) × 6 • •
24 2×4 = . 168 (1 + 6) × 8 •
•
27 2+7 = . 168 (1 + 6) × 8 35 3×5 = . 189 (1 + 8) × 9
•
3×6 36 = . 198 1 + 98 37 3+7 = . 148 (1 + 4) × 8
4+2 42 = . • 315 3 × 15 42 4+2 • = . 385 (3 + 8) × 5 42 4+2 • = . 735 7×3×5
135 1×3×5 = . 648 6 × (4 + 8)
•
1×3×5 135 = . 729 72 + 9 •
• •
135 1×3×5 = . 468 4+6×8
219 (2 + 1) × 9 • = . 365 (3 + 6) × 5
• •
•
135 1 × 35 = . 297 (2 + 9) × 7
136 (1 + 3) × 6 = . 459 (4 + 5) × 9 •
•
•
•
•
14 + 7 147 = . 385 (3 + 8) × 5 153 1 + 53 = . 476 4×7×6 15 + 3 153 = . 748 (7 + 4) × 8 154 1+5+4 = . 693 (6 + 9) × 3
•
•
•
•
235 (2 + 3) × 5 = . 846 84 + 6 235 (2 + 3) × 5 = . 987 98 + 7 261 2×6×1 = . 957 9+5×7 273 2×7×3 = . 468 4 + 68 2 × (7 + 8) 278 = . 695 (6 + 9) × 5 286 2+8+6 = . 715 (7 + 1) × 5
157 (1 + 5) × 7 = . 628 6 × 28
•
157 1×5+7 = . 942 9×4×2
•
318 (3 + 1) × 8 = . 795 (7 + 9) × 5
•
14 1+4 = . 9576 9 × 5 × 76
•
158 (1 + 5) × 8 = . 632 6 × 32
•
(3 + 4) × 5 345 = . 621 62 + 1
•
17 1+7 = . 3468 34 × 6 × 8
84 8+4 = . 315 3 × 15
•
168 (1 + 6) × 8 = . 297 2 + 97
•
351 3 + 51 = . 728 7×2×8
•
17 1+7 = . 4386 43 × 8 × 6
84 8+4 = . 735 7×3×5
•
187 (1 + 8) × 7 = . 935 9 × 35
•
357 35 + 7 = . 612 6 × 12
•
17 1+7 = . 4692 4 × 6 × 92
•
51 5+1 = . 748 (7 + 4) × 8
•
•
71 7+1 = . 426 4×2×6
•
•
78 7+8 = . 156 1×5×6
•
•
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
21 2×1 = . 3675 (3 + 67) × 5
•
86 8+6 = . 1935 1 × 9 × 35
•
192 (1 + 9) × 2 = . 3456 3×4×5×6
•
268 2+6+8 = . 3417 3 × 4 × 17
•
21 2+1 = . 6384 6 × 38 × 4
•
97 9+7 = . 4365 4 × 36 × 5
•
192 1+9+2 = . 5376 (5 + 3) × 7 × 6
•
286 2×8+6 = . 1495 (14 + 9) × 5
•
28 2+8 = . 4536 45 × 36
•
123 (1 + 2) × 3 = . 4879 4 × 87 + 9
•
193 1×9+3 = . 4825 (4 + 8) × 25
•
289 2 × (8 + 9) = . 1734 17 × 3 × 4
28 2+8 • = . 9576 9 × 5 × 76
•
124 1+2×4 = . 7936 (7 + 9) × 36
•
291 2+9+1 = . 4365 4 × (3 + 6) × 5
39 3+9 • = . 1248 12 × 4 × 8
•
1+2+6 126 = . 3584 (3 + 5) × 8 × 4
•
291 2×9×1 = . 8536 (85 + 3) × 6
134 1×3×4 = . 5896 58 × 9 + 6
•
•
297 29 + 7 = . 1584 (1 + 5) × 8 × 4
•
297 29 + 7 = . 1683 1 × 68 × 3
•
297 2+9+7 = . 3465 (3 + 4) × 6×)
2×1×7 217 = . 9486 (94 + 8) × 6
•
2+9+7 297 = . 8316 (83 + 1) × 6
•
23 + 1 231 = . 5698 (5 + 69) × 8
•
316 (3 + 1) × 6 = . 4582 4 × (5 + 82)
•
235 (2 + 3) × 5 = . 7896 7 × 8 × (9 + 6)
•
(3 + 1) × 8 318 = . 5724 572 + 4
•
241 2+4×1 = . 3856 (3 + 8 + 5) × 6
•
319 3+1×9 = . 4785 (4 × 7 + 8) × 5
•
243 24 + 3 = . 9576 (9 + 5) × 76
•
324 32 + 4 = . 9576 (9 + 5) × 76
247 2+4+7 • = . 1539 (1 + 5 + 3) × 9
•
(3 + 2) × 5 325 = . 9841 9 × 84 + 1
•
329 3+2×9 = . 6815 (6 + 81) × 5
•
342 3 × (4 + 2) = . 1596 (1 × 5 + 9) × 6
•
349 3+4+9 = . 8725 (8 + 72) × 5
•
352 (3 + 5) × 2 = . 1496 14 + 9 × 6
261 2×6×1 = . 5394 (53 + 9) × 4
•
352 (3 + 5) × 2 = . 8976 8 × (9 + 7 × 6)
•
357 (3 + 5) × 7 = . 1428 14 × 2 × 8
•
357 (3 + 5) × 7 = . 4182 41 × 8 × 2
•
•
•
3+9 39 = . 1768 (1 + 7) × 68 39 3+9 = . 2184 21 × 8 × 4 5+1 51 = . 4692 4 × 69 × 2
51 5×1 • = . 4896 4 × 8 × (9 + 6)
•
5+3 53 = . 2968 (2 + 9 × 6) × 8
136 (1 + 3) × 6 • = . 2754 (2 + 7) × 54 138 1+3+8 • = . 5796 (5 + 79) × 6
•
•
• •
57 5+7 = . 1368 1 × 36 × 8 •
•
5+7 57 = . 2698 (2 + 69) × 8 •
•
•
•
•
62 6+2 = . 1395 (1 + 3) × (9 × 5) 63 6+3 = . 4872 4 × 87 × 2 78 7+8 = . 1456 (1 + 4) × 56 81 8+1 = . 9576 (9 + 5) × 76
•
138 1+3+8 = . 7245 7 × 2 × 45
195 1+9+5 • = . 3276 3×2×7×6 195 1+9+5 • = . 4368 (4 + 3) × 6 × 8
•
•
•
149 1+4+9 = . 3576 (3 + 5) × 7 × 6 149 1+4×9 = . 3725 37 × 25 15 + 3 153 = . 4692 4 × 69 × 2 157 (1 + 5) × 7 = . 4239 42 × 3 × 9 176 1×7+6 = . 4928 (4 + 9) × 28
179 1+7×9 = . • 2864 2 × 8 × 64 179 1×7+9 • = . 8234 8 × 23 × 4
•
•
213 (2 + 1) × 3 = . 6958 6 × (9 + 5 × 8) 217 2+1×7 = . 3689 (3 + 6 + 8) × 9
247 2+4+7 = . 1596 (1 × 5 + 9) × 6 (2 + 4) × 8 248 = . 9765 9×7×6×5 254 (2 + 5) × 4 = . 1397 (13 + 9) × 7
•
182 (1 + 8) × 2 = . 5369 (53 + 6) × 9
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82 8+2 = . 1435 (1 + 4) × 35
•
184 (1 + 8) × 4 = . 3726 3 + 726
•
•
82 8×2 = . 1476 (1 + 47) × 6
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187 1 + 87 = . 4692 4 × 6 × 92
•
264 2×6+4 = . 3795 (37 + 9) × 5
•
84 8+4 = . 1365 1 × 3 × 65
•
1+8+9 189 = . 3675 (3 + 67) × 5
•
2×6×5 265 = . 1378 (1 + 3) × 78
•
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
357 35 + 7 = . 4692 4 × 69 × 2
•
425 4 × 25 = . 3978 (3 + 9) × 78
•
572 (5 + 7) × 2 = . 9438 9 × 4 × (3 + 8)
•
358 (3 + 5) × 8 = . 4296 4 × 2 × 96
•
429 4 × (2 + 9) = . 1638 (1 + 6) × 3 × 8
•
574 5×7×4 = . 1968 (1 + 9) × (6 × 8)
364 (3 + 6) × 4 • = . 1729 (17 + 2) × 9
429 4+2×9 • = . 6318 6 × 3 × 18
364 (3 + 6) × 4 • = . 7189 (71 + 8) × 9
429 4+2+9 • = . 7865 (7 + 8 × 6) × 5
(3 + 6) × 8 368 = . 1472 1 × 4 × 72
•
•
374 3 × (7 + 4) = . 2856 (2 + 8 × 5) × 6
•
•
•
•
729 72 + 9 = . 1485 (1 + 4 × 8) × 5
•
7×3×2 732 = . 1586 1 × 5 + 86
•
738 7+3+8 = . 4592 4 × (5 + 9) × 2
•
748 (7 + 4) × 8 = . 1326 13 × 2 × 6
594 (5 + 9) × 4 = . 7128 7 × 12 × 8
457 (4 + 5) × 7 = . 6398 (6 + 3) × 98
•
6+1+2 612 = . 5984 (5 + 9 + 8) × 4
•
748 (7 + 4) × 8 = . 1632 1 × 6 × 32
374 3 × (7 + 4) = . 5168 (51 + 6) × 8
•
459 45 + 9 = . 1326 13 × 2 × 6
•
612 6×1×2 = . 7548 7×5×4+8
•
748 (7 + 4) × 8 = . 3162 31 × 6 × 2
3 × (8 + 4) 384 = . 2176 2 × 17 × 6
•
459 45 + 9 = . 3162 31 × 6 × 2
6 + 27 627 = . 1539 (1 + 5 + 3) × 9
•
•
7×4+8 748 = . 5236 (5 + 2) × 36
•
754 7×5+4 = . 1392 (1 + 3) × 9 × 2
•
756 7 + 56 = . 4128 (41 + 2) × 8
•
783 78 + 3 = . 1624 (1 + 6) × 24
(3 + 9) × 2 392 = . 6517 (6 + 51) × 7
462 4+6+2 • = . 7315 (7 + 31) × 5
•
•
394 3+9+4 = . 2758 (2 + 7 + 5) × 8
•
•
4+6×8 468 = . 1593 1 × 59 × 3
594 5+9+4 = . 8613 (86 + 1) × 3
6 + 29 629 = . • 7548 7 × 5 × (4 + 8)
•
635 (6 + 3) × 5 = . 9271 9 × (2 + 71)
476 4+7+6 = . 3192 3 × 19 × 2
•
639 6+3+9 = . 2485 (2 + 4 + 8) × 5
•
785 (7 + 8) × 5 = . 4239 (42 + 3) × 9
•
(4 + 8) × 3 483 = . 2576 (25 + 7) × 6
•
6 × (4 + 8) 648 = . 1593 1 × 59 × 3
•
791 7×9×1 = . 2486 24 × 8 + 6
395 (3 + 9) × 5 = . 1264 (1 + 2) × 64
•
486 48 + 6 = . 1593 1 × 59 × 3
•
651 6×5×1 = . 2387 23 + 87
•
792 (7 + 9) × 2 = . 1386 (1 + 3) × (8 + 6)
396 (3 × 9) + 6 = . 2784 (2 + 7 × 8) × 4
•
(4 + 9) × 5 495 = . 1386 13 × (8 + 6)
693 (6 + 9) × 3 • = . 8547 8 + 547
•
792 (7 + 9) × 2 = . 1485 (1 × 4 + 8) × 5
•
516 (5 + 1) × 6 = . 3827 3 × (82 + 7)
694 6+9×4 • = . 1735 1×7×3×5
•
792 7×9×2 = . 3564 3 + 564
•
792 7 × (9 + 2) = . 5184 (5 + 1) × 84
•
792 (7 + 9) × 2 = . 8514 (85 + 1) × 4
•
795 (7 + 9) × 5 = . 3816 3 × 8 × 16
•
816 8×1×6 = . 3927 3 × (9 + 2) × 7
•
8×1×6 816 = . 7259 7 × (2 + 59)
3×9+6 396 = . • 4128 (41 + 2) × 8 412 (4 + 1) × 2 = . 6798 67 + 98
529 5+2×9 • = . 3864 3 × (8 + 6) × 4
•
6 × (9 + 7) 697 = . 3485 3×4×8×5 714 7×1×4 = . 3825 3 × (8 + 2) × 5
529 (5 + 2) × 9 = . 6348 63 × (4 + 8)
•
•
546 (5 + 4) × 6 = . 1729 (17 + 2) × 9
•
728 (7 + 2) × 8 = . 3549 (35 + 4) × 9
•
(5 + 4) × 6 546 = . 7189 (71 + 8) × 9
•
(7 + 2) × 8 728 = . 5369 (53 + 6) × 9
415 4+1+5 = . 6723 6 × (7 + 2) × 3
•
•
415 (4 + 1) × 5 = . 8632 8 × (63 + 2)
•
4 + 23 423 = . 6815 (6 + 81) × 5
•
729 7 + 29 = . 1458 1 × (4 + 5) × 8
•
•
•
•
•
392 3+9×2 = . 7168 (7 + 1) × 6 × 8
•
•
729 72 + 9 = . 1368 (1 + 3 × 6) × 8
436 4 × (3 + 6) = . 8175 (8 + 1) × 75
384 3 × (8 + 4) • = . 6912 6 × 9 × 12
•
581 5+8+1 • = . 6723 6 × (7 + 2) × 3
•
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
823 (8 + 2) × 3 = . 5761 5×7×6×1
•
942 9 × (4 + 2) = . 7536 (7 + 5) × 36
•
48 4+8 = . 92736 92 × 7 × 36
•
1386 1 × 38 + 6 = . 4725 (4 × 7 + 2) × 5
•
832 (8 + 3) × 2 = . 4576 45 + 76
•
947 9+4+7 = . 8523 (8 + 52) × 3
•
51 5+1 = . 34986 (3 + 4) × 98 × 6
•
1392 (1 + 39) × 2 = . 4785 (47 + 8) × 5
•
837 8×3+7 = . 2916 2×9×1×6
•
9×5×1 951 = . 3487 3 × (48 + 7)
•
54 5+4 = . 37296 3 × 7 × 296
•
1396 (1 + 3) × (9 + 6) = . 8725 (8 + 7) × 25
•
8+4×2 842 = . 6315 6 × (3 + 1) × 5
•
952 (9 + 5) × 2 = . 1734 17 + 34
•
6+2 62 = . 13485 1 × 348 × 5
•
1+3×9+8 1398 = . 6524 6 × (5 + 2) × 4
•
952 (9 + 5) × 2 = . 1836 18 + 36
62 6×2 • = . 19375 (1 + 9) × 375
•
1425 1 × 4 × 25 = . 3876 38 × 7 + 6
•
1458 1 + 45 + 8 = . 3726 (3 × 7 + 2) × 6
•
1458 1+4+5+8 = . 6237 (6 + 2 + 3) × 7
•
1472 (1 × 4 + 7) × 2 = . 9568 95 + 6 × 8
•
(1 + 4 + 7) × 6 1476 = . 2583 (2 + 5 × 8) × 3
845 (8 + 4) × 5 • = . 9126 9 × 12 × 6 8+4+6 846 = . • 3572 (3 + 5 × 7) × 2
•
•
•
•
854 8+5×4 = . 2196 (2 + 1 + 9) × 6 879 8+7+9 = . 3516 (3 × 5 + 1) × 6 891 8 + 91 = . 3645 (3 + 6) × 45 8+9+1 891 = . 4752 4 × (7 + 5) × 2
952 (9 + 5) × 2 • = . 3468 34 + 68 952 (9 + 5) × 2 • = . 3876 38 + 76 952 (9 + 5) × 2 • = . 4386 43 + 86
891 8+9+1 = . 6534 (6 + 5) × 3 × 4 8+9+1 891 = . 7425 (7 × 4 + 2) × 5 •
918 9 + 18 • = . 3264 3 × (2 + 6) × 4 918 9+1+8 • = . 6375 (6 × 3 + 7) × 5
•
927 9 + 27 = . 4635 4 × (6 + 3) × 5
•
•
•
7+1 71 = . 38695 (3 + 869) × 5
•
1482 14 × 8 + 2 = . 7956 (7 + 95) × 6
•
72 7+2 = . 34816 34 × 8 × 16
•
1485 1+4+8×5 = . 2376 (2 + 3 + 7) × 6
•
8+1 81 = . 59472 (5 + 9) × 472
•
1485 (1 × 4 + 8) × 5 = . 6237 6×2×3×7
•
83 8×3 = . 26975 (2 + 6) × 975
•
1495 (14 + 9) × 5 = . 3276 3×2×7×6
•
(1 × 4 + 9) × 5 1495 = . 3726 3 × (7 + 2) × 6
•
1568 (1 + 5 + 6) × 8 = . 3724 3 × (72 + 4)
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1584 1×5×8×4 = . 2673 267 + 3
•
1584 (1 + 5) × 8 × 4 = . 3762 3 × 76 × 2
•
1584 1×5×8×4 = . 6237 623 + 7
986 (9 + 8) × 6 = . 3451 (3 + 4) × 51 1+3 13 = . 56784 56 × 78 × 4
2×1 21 = . 35679 3 + 5 × 679
71 7+1 = . 25986 (2 + 59) × 8 × 6
7+2 72 = . 16384 (1 + 63) × 8 × 4
986 (9 + 8) × 6 = . 2175 (2 + 1) × 75
18 1+8 = . 79632 7 × 9 × 632
71 7+1 = . 23856 2 × 3 × 8 × 56
• 967 9+6×7 • = . 4835 (48 + 3) × 5
• •
•
(9 + 5) × 3 953 = . • 7624 7×6×2×4
• •
•
84 8+4 • = . 92736 9 × 2 × 736 91 9×1 • = . 38675 (3 + 8 × 6) × 75
34 3+4 • = . 19856 (1 + 9 × 8) × 56
•
93 9×3 = . 16275 (1 + 62) × 75 1265 1×2×6×5 = . 8349 (8 + 3) × 4 × 9
931 9×3+1 = . 8645 (8 × 6 + 4) × 5
•
37 3+7 = . 15984 15 × 9 × 8 × 4
•
•
936 (9 + 3) × 6 = . 1287 12 + 87
•
39 3+9 = . 56784 56 × 78 × 4
•
1287 (1 + 2 + 8) × 7 = . 4563 45 × 6 + 3
•
1586 (1 × 5 + 8) × 6 = . 4392 4×3×9×2
•
936 (9 + 3) × 6 = . 1482 14 × 8 + 2
•
43 4+3 = . 16985 (1 + 69 × 8) × 5
•
1386 1 × 3 × (8 + 6) = . 2475 (2 × 4 + 7) × 5
•
1634 1 + 6 × (3 + 4) = . 9728 (9 + 7) × 2 × 8
•
9+3×6 936 = . 5824 (5 × 8 + 2) × 4
•
4×3 43 = . 18576 (1 + 8) × 576
•
1386 (13 + 8) × 6 = . 2794 2+7×9×4
•
16 × (7 + 2) 1672 = . 4389 (4 + 38) × 9
•
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
1683 1×6×8+3 = . 2475 (2 × 4 + 7) × 5
•
1938 1+9+3×8 = . 4275 (4 × 2 + 7) × 5
•
2475 (2 × 4 + 7) × 5 = . 8613 (86 + 1) × 3
•
2987 29 + 87 = . 4635 4 × (6 + 3) × 5
•
1683 1×6+8+3 = . 7425 (7 + 4 × 2) × 5
•
1938 1 + 93 + 8 = . 6745 (67 + 4) × 5
•
248 (2 + 4) × 8 = . 19375 (1 + 9) × 375
•
3124 (31 + 2) × 4 = . 6958 6 × (9 + 5 × 8)
•
1683 (1 + 6 + 8) × 3 = . 9724 (9 × 7 + 2) × 4
•
1963 19 × 6 + 3 = . 7248 (7 + 2) × 48
•
2574 2 × (5 × 7 + 4) = . 8613 (86 + 1) × 3
•
(3 + 1) × 6 + 8 3168 = . 7425 (7 + 4 × 2) × 5
•
3192 3 + 1 + 92 = . 8645 (8 × 6 + 4) × 5
•
3249 3 + (2 + 4) × 9 = . 5187 (5 + 1 × 8) × 7
•
3 + 25 + 8 3258 = . 7964 (7 + 9 + 6) × 4
•
3267 3 × (2 + 6 × 7) = . 8514 (85 + 1) × 4
•
3276 3×2×7×6 = . 9841 9 × 84 + 1
2 × (76 + 1) 2761 = . 9538 (9 + 5) × 38
•
3 + 41 + 7 3417 = . 6298 6 + (2 + 9) × 8
(1 + 7 × 2) × 5 1725 = . • 3864 3 × (8 + 6) × 4
•
•
1725 1 × 7 × 25 = . 4968 4 × 9 × (6 + 8) 1728 1+7+2×8 = . 4536 4 + 53 + 6
(1 + 9 + 7) × 2 1972 = . • 3654 3 + 6 + 54
•
•
1729 1 + 7 × (2 + 9) = . 6384 (6 + 3) × 8 × 4
•
•
1732 (1 + 7 × 3) × 2 = . 6495 (6 × 4 + 9) × 5
•
•
1749 (1 × 7 + 4) × 9 = . 3286 (3 + 28) × 6
•
•
1976 (19 + 7) × 6 = . 2584 25 × 8 + 4 2136 2 × (1 + 3) × 6 = . 5874 58 + 74 (2 + 1 + 4) × 5 2145 = . 9867 (9 + 8 + 6) × 7
2583 (2 + 5 + 8) × 3 • = . 9471 94 + 71 •
•
•
(2 × 5 + 8) × 4 2584 = . 9367 9 + 36 × 7 2598 2 × 5 + 98 = . 4763 (4 + 7) × 6 × 3 2673 26 + 73 = . 5184 (5 + 1) × 8 × 4
2175 (2 + 1 + 7) × 5 = . 4698 4 + 6 + 98
•
•
2175 2 × 1 × 75 = . 6438 6 + 438
•
2784 (2 + 7) × 8 × 4 = . 3596 (3 + 59) × 6
•
3465 3 × (4 + 6) + 5 = . 7128 (7 + 1 × 2) × 8
(1 × 7 + 5) × 2 1752 = . 3869 38 + 6 + 9
•
2178 2×1×7+8 = . 6435 (6 + 4 + 3) × 5
•
2784 (2 + 7) × 8 × 4 = . 3915 3 × 9 × 15
•
(3 + 46) × 5 3465 = . 8217 (82 + 1) × 7
•
1764 (1 + 7 + 6) × 4 = . 2835 2 + 83 + 5
•
2184 2 × 1 × 84 = . 7956 (7 + 95) × 6
•
2793 2 + 79 + 3 = . 8645 (8 × 6 + 4) × 5
•
3479 3 + 47 × 9 = . 6125 6 × 125
•
1782 (1 + 7) × 8 + 2 = . 3645 (3 + 6 × 4) × 5
•
2185 (2 + 1 + 8) × 5 = . 3496 34 + 9 × 6
•
2813 (28 + 1) × 3 = . 7469 7 × (4 × 6 + 9)
•
(35 + 2) × 8 3528 = . 6174 (6 + 1) × 74
1782 1+7+8+2 = . • 6435 (6 + 4 + 3) × 5
•
2195 2 × (1 + 9 + 5) = . 7463 (7 × 4 + 6) × 3
2834 2 × (8 + 3) + 4 = . 9156 (9 + 1 × 5) × 6
•
•
3542 (3 + 5) × 42 = . 9867 9 × 8 × (6 + 7)
•
3564 3 + 5 + 64 = . 8712 (87 + 1) × 2
•
3 + 67 + 2 3672 = . 8415 (8 × 4 + 1) × 5
•
3726 (3 × 7 + 2) × 6 = . 5184 (5 + 1) × 8 × 4
•
3915 3×9×1×5 = . 4872 (4 + 8) × 7 × 2
•
3925 (3 + 9) × 25 = . 4867 4 × (86 + 7)
•
4+2+7×8 4278 = . 9315 9×3×1×5
•
4536 4×5×3×6 = . 7182 71 × 8 + 2
•
•
1794 1 + 7 × (9 + 4) = . 6825 (68 + 2) × 5 1827 182 + 7 = . 3596 (3 + 59) × 6
2317 (2 × 3 + 1) × 7 = . • 4965 4 + 96 + 5
•
1846 1+8×4+6 = . 7952 (79 + 5) × 2
•
1 × 8 + 56 1856 = . 2349 (2 + 3 + 4) × 9
•
•
1863 1×8+6×3 = . 7452 (7 + 45) × 2
•
1876 (1 + 8 + 7) × 6 = . 2345 2×3×4×5
•
•
2376 2 × (3 + 7) × 6 = . 8415 (84 + 1) × 5 2394 2×3+9×4 = . 5187 (5 + 1 × 8) × 7
2835 2 + 83 + 5 • = . 6174 (6 + 1) × 7 × 4
•
•
2845 (2 + 8) × 4 + 5 = . 9673 (9 + 6 × 7) × 3 2853 28 + 5 + 3 = . 6974 (6 + 9 + 7) × 4
2431 2 × 4 + 31 = . 8976 (8 + 9 + 7) × 6
•
•
2438 2 + 43 + 8 = . 5796 (5 + 7 + 9) × 6
•
2871 2 × 87 × 1 = . 3465 (3 + 4) × 6 × 5
•
(2 + 4 × 7) × 5 2475 = . 3168 (3 + 1) × 6 × 8
•
2+8×7×1 2871 = . 6534 (6 + 5) × 3 × 4
2869 2 × (8 × 6 + 9) = . 5134 (5 + 1) × 34
Inder J. Taneja
•
5984 (5 + 9 + 8) × 4 = . 6732 67 + 32
•
4653 4 × (6 + 5) + 3 • = . 7128 (7 + 1 × 2) × 8
•
6237 6×2×3×7 = . 8514 (85 + 1) × 4
143 1×4×3 • = . 26598 (26 + 5) × 9 × 8
4653 4+6×5×3 • = . 8712 (87 + 1) × 2
6 × (31 + 8) 6318 = . • 7425 (7 + 4) × 25
•
4627 (4 + 6 + 2) × 7 = . 8593 (8 + 5) × (9 + 3)
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
259 2+5+9 = . 14763 1 × 4 × 76 × 3
•
259 2+5+9 = . 47138 4 × 7 × 13 × 8
•
261 2+6×1 = . 37584 3 × (7 + 5) × 8 × 4
157 1×5+7 = . 23864 2 × 38 × 6 × 4
•
2+6+1 261 = . 37845 (3 × 7 + 8) × 45
•
273 27 + 3 = . 18564 1 × 85 × 6 × 4
•
279 27 + 9 = . 13485 1 × 348 × 5
•
296 2+9×6 = . 13875 1 × 3 × 875
•
3×1×5 315 = . 28476 (2 + 8 × 4 × 7) × 6
•
316 3 × 16 = . 45978 (4 + 5) × 97 × 8
•
317 31 + 7 = . 58962 589 × 6 × 2
•
324 3 × (2 + 4) = . 71568 7 × 1 × 568
148 1×4+8 • = . 69375 (6 + 9) × 375
•
6534 6+5×3×4 = . 7128 (7 + 1 × 2) × 8
•
16 + 2 162 = . 58473 (5 + 84) × 73
(4 + 8) × 7 × 2 4872 = . 9135 9 × 1 × 35
•
7152 (7 + 1) × 5 × 2 = . 9834 98 + 3 × 4
•
168 1+6+8 = . 74592 74 × 5 × 9 × 2
4893 (4 + 8 + 9) × 3 = . 5126 5 × 12 + 6
•
(7 + 2) × 3 × 6 7236 = . 9514 9 + 51 × 4
•
172 1×7+2 = . 93568 9 × (3 + 5) × 68
•
•
4732 (4 + 7 + 3) × 2 = . 6591 6×5+9×1
•
•
4896 (4 + 8 + 9) × 6 • = . 5712 5 + 71 × 2 (4 + 92) × 3 4923 = . 7658 7 × (6 + 58)
(7 + 3 + 1) × 5 7315 = . • 8246 8 × 2 + 46
•
4972 4 + 97 × 2 = . 8136 (8 + 1) × 36
•
•
4981 4 × (9 + 8 × 1) = . 7325 (7 + 3) × 2 × 5
•
•
7425 (7 × 4 + 2) × 5 = . 8316 8 × 3 × (1 + 6)
153 1+5+3 = . 29648 (2 + 9 × 6 × 4) × 8
1+7×4 174 = . • 29568 (2 + 9) × 56 × 8
•
193 (1 + 9) × 3 = . 56742 5 × 6 × 7 × 42 195 1+9+5 = . 48672 (4 + 8 × 6) × 72
(7 + 4) × 3 + 6 7436 = . 9152 (9 + 15) × 2
•
•
7684 7×6×8+4 = . 9153 9 × 15 × 3
•
214 2+1×4 = . 37985 3 × (7 × 9 + 8) × 5
•
3×2×4 324 = . 81756 8 × (1 + 756)
5148 (5 + 1 × 4) × 8 = . 7293 7 + 2 + 93
•
8432 (8 + 43) × 2 = . 9176 9 + 17 × 6
•
231 2×3+1 = . 47685 (4 + 7 + 6) × 85
•
346 (3 + 4) × 6 = . 12975 (12 + 9) × 75
5 + 14 × 8 5148 = . 7392 7 × (3 + 9) × 2
•
(8 × 7 + 3) × 2 8732 = . 9546 9+5×4×6
•
231 2×3+1 = . 48675 (48 × 6 + 7) × 5
•
349 (3 + 4) × 9 = . 26175 (2 + 61) × 75
2×4+1 241 = . • 39765 (3 × 97 + 6) × 5
•
(3 + 5) × 2 352 = . 16984 16 + 9 × 84
•
367 3+6+7 = . 12845 1 × 28 × 4 × 5
•
369 3 + 69 = . 17425 (1 + 7) × 425
243 2+4+3 = . 15876 (1 + 5 + 8) × 7 × 6
•
372 3+7+2 = . 19685 (1 + 9 × (6 + 8)) × 5
•
374 3 + 74 = . 19856 (1 + 9 × 8) × 56
•
3+7+8 378 = . 15246 (1 + 5 × 24) × 6
5312 (5 + 3 × 1) × 2 • = . 8964 8+9+6+4 5319 (5 + 3 + 1) × 9 = . 7486 (7 + 4 + 8) × 6
129 1+2+9 • = . 48375 (4 + 8) × 375
•
5319 5 × (3 + 1) × 9 = . 8274 (8 + 2) × 7 × 4
•
•
5412 5 × 41 × 2 = . 8976 8 × (9 + 76)
•
56 + 43 5643 = . 7182 7 × (1 + 8) × 2
•
2×5+1 251 = . 36897 3 × (68 + 9) × 7
•
4728 4 × (7 + 2) × 8 = . 9653 9 × 65 + 3
•
•
6435 (6 + 4 + 3) × 5 = . 7128 (7 + 1 × 2) × 8
•
•
246 24 + 6 = . 31857 3 × 185 × 7
•
•
•
•
6419 6+4×1×9 = . 8253 (8 + 2 × 5) × 3
4718 47 + 1 + 8 = . 5392 (5 + 3 × 9) × 2
•
142 1×4+2 = . 57936 (5 + 7 × 9) × 36
132 (1 + 3) × 2 = . 47685 (4 × 7 + 6) × 85
•
243 2+4+3 = . 15687 (15 + 68) × 7
132 1+3+2 = . 48576 4 × (85 + 7) × 6
•
•
135 1×3×5 = . 27864 (2 + 7) × 86 × 4
•
243 2+4+3 = . 75168 (7 + 51) × 6 × 8
•
14 + 2 142 = . 38695 (3 + 869) × 5
•
(2 + 4) × 3 243 = . 81756 8 × (1 + 756)
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
378 3+7+8 = . 15624 (1 + 5 × 6) × 24
•
471 47 + 1 = . 68295 6 × 8 × 29 × 5
•
567 5+6+7 = . 41328 4 × 1 × 328
•
742 7 × (4 + 2) = . 13568 (1 + 3 × 5) × 6 × 8
•
378 3+7+8 = . 19425 (1 + 9 × 4) × 25
•
473 4 + 73 = . 16985 (1 + 69 × 8) × 5
•
576 5×7+6 = . 13824 1 × 3 × 82 × 4
•
742 7 + 42 = . 39856 (39 + 8) × 56
•
752 (7 + 5) × 2 = . 69184 69 × 1 × 8 × 4
•
753 (7 + 5) × 3 = . 42168 42 × 1 × 6 × 8
•
7 + 56 756 = . 23184 23 × 1 × 84
•
756 7 + 56 = . 41328 41 × 3 × 28
•
758 (7 + 5) × 8 = . 39416 (3 + 9) × 416
379 3 × (7 + 9) • = . 48512 (4 + 8) × 512
•
381 3×8×1 = . 24765 24 × (7 + 6) × 5
478 4×7+8 • = . 15296 (1 + 5) × 2 × 96
•
•
(4 + 8) × 2 482 = . 39765 (3 + 9 × 7) × 6 × 5
•
612 6×1×2 = . 34578 3 + 45 × (7 + 8)
4+1×7 417 = . 93825 9 × (3 + 8) × 25
•
491 4×9+1 = . 23568 (2 + 35) × 6 × 8
•
612 6+1+2 = . 73984 (7 + 3 × 9) × 8 × 4
•
7×5+9 759 = . 13248 (1 + 3) × 24 × 8
4+1×8 418 = . 67925 6 × (7 × 9 + 2) × 5
•
491 4+9+1 = . 36825 3 × (68 + 2) × 5
•
621 6 × (2 + 1) = . 75348 7 × (5 + 34) × 8
•
759 7 + 59 = . 23184 (23 + 1) × 84
419 4×1×9 = . 36872 (36 + 8) × 72
497 49 + 7 • = . 23856 2 × 3 × 8 × 56
6×2×4 624 = . • 15873 (1 + 58 × 7) × 3
•
762 7×6×2 = . 14859 14 × (8 + 5) × 9
•
786 7 × (8 + 6) = . 49125 49 × 125
•
792 7 + 92 = . 16384 (1 + 63) × 8 × 4
•
792 7 + 92 = . 34816 34 × 8 × 16
•
•
387 3+8+7 = . 16254 (1 + 6) × 2 × 54
•
•
423 (4 + 2) × 3 = . 56917 (5 × 69 + 1) × 7
•
•
4 + 29 429 = . 15873 (1 + 58 × 7) × 3
•
•
429 4 + 29 = . 75816 (7 + 5) × 81 × 6
•
•
•
•
624 6×2+4 = . 85137 (8 + 51) × 37
•
•
519 5 × 19 = . 23874 (2 + 3) × 874
•
6 + 49 649 = . 13275 (1 + 32 × 7) × 5
•
7+9+8 798 = . 21546 2 × 1 × 54 × 6
431 4×3×1 = . 56892 (5 + 6) × 8 × 9 × 2
•
523 5×2×3 = . 19874 19 × (8 + 7) × 4
•
652 6+5×2 = . 31948 (3 + 1 + 94) × 8
•
798 7+9+8 = . 23541 2 × 354 × 1
432 4×3×2 = . 15876 1 + 5 + 876
524 (5 + 2) × 4 • = . 79386 7 × (93 + 8) × 6
682 6+8×2 = . • 14973 (14 + 9) × 7 × 3
•
814 8×1×4 = . 79365 (7 + 9) × 3 × 65
•
4+3+2 432 = . 17856 (1 + 7 × 8 + 5) × 6
528 5 + 28 • = . 14976 (149 + 7) × 6
816 8 + 16 = . 43792 (4 + 3 + 7) × 92
•
849 8+4+9 = . 12735 1 × (2 + 7) × 35
•
872 (8 + 7) × 2 = . 54936 5 × (4 × 93 + 6)
•
873 8 + 73 = . 14259 (142 + 5) × 9
•
913 9 + 13 = . 68475 6 × (8 + 47) × 5
•
9+1+8 918 = . 24735 (2 × 47 + 3) × 5
•
627 6 + 27 = . 41895 (41 + 8) × 9 × 5
689 6+8×9 = . 72345 7 × 234 × 5 693 6+9+3 = . 17248 1×7×2×4×8
532 5 × 32 = . 14896 (1 + 4) × 896
•
543 5+4×3 = . 79821 7 × (9 + 8) × 21
•
•
562 5+6×2 = . 17984 (1 + 7 + 9) × 8 × 4
•
721 72 × 1 = . 68495 6 × (8 + 4) × 95
•
567 5+6+7 = . 23184 23 × 1 × 8 × 4
•
(7 + 3) × 6 736 = . 15824 15 × (82 + 4)
•
451 45 × 1 = . 36982 3 × (6 + 9) × 82
•
•
453 4 + 53 = . 12986 (1 + 2 × 9) × 86
•
457 4+5×7 = . 21936 2 × 1 × 936
•
498 4+9×8 = . 23157 2 × 31 × 57 514 5 × 14 = . 32896 (3 + 2) × 896
439 4 × (3 + 9) = . 76825 7 × 6 × 8 × 25
•
594 59 + 4 = . 13728 13 × 7 × 2 × 8
•
3+8+1 381 = . 75946 (7 + 5 × 9) × 46
•
5+9+1 591 = . • 37824 3 × (78 + 2) × 4
482 48 × 2 = . 37596 (3 + 75) × 96
•
•
481 4+8×1 = . 79365 (7 × 9 + 3) × 6 × 5
576 (5 + 7) × 6 • = . 18432 18 × 4 × 32
7×2×1 721 = . 43569 (4 + 3 × 5 × 6) × 9
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
918 9×1×8 = . 47532 4 + 7 × 532
•
39 3+9 = . 184275 18 × 42 × 75
•
1836 1 × 8 × (3 + 6) = . 49725 (4 × 97 + 2) × 5
•
2489 2 × (48 + 9) = . 16375 (1 + 6 + 3) × 75
•
924 (9 + 2) × 4 = . 18375 (1 + 8 × 3) × 7 × 5
•
42 4+2 = . 197568 1 × 9 × 7 × 56 × 8
•
1836 1+8+3+6 = . 95472 9 × (5 + 47) × 2
•
2513 2 × 51 + 3 = . 97648 (9 + 76) × 48
•
9+2+7 927 = . 53148 (5 + 31 × 4) × 8
•
4+6 46 = . 129375 (1 + 2) × 9375
•
1837 1×8+3+7 = . 64295 (6 + 4 × 2) × 9 × 5
•
2516 2 + 5 × (1 + 6) = . 73984 (7 + 3 × 9) × 8 × 4
79 7+9 = . 138645 13 × 8 × 6 × 45
1845 1 × 8 × 45 • = . 37269 3 + 7269
•
•
2576 (2 + 5) × 7 × 6 = . 13984 (1 + 398) × 4
•
2593 25 + 9 × 3 = . 67418 (6 × 7 × 4 + 1) × 8
•
2613 2 × 6 × 13 = . 97485 97 × (4 + 8) × 5
•
2754 27 × (5 + 4) = . 39168 3 × 9 × 16 × 8
•
2793 2+7+9+3 = . 61845 (61 + 8 × 4) × 5
934 9+3+4 • = . 81725 8 × 1 × 7 × 25
•
•
936 (9 + 3) × 6 = . 54782 54 × 78 + 2 947 9+4+7 = . 13258 (1 + 3 × 2) × 5 × 8
12 + 36 1236 = . • 84975 (8 + 4 × 9) × 75
•
1274 1+2+7+4 = . 95368 (95 + 36) × 8
•
•
1×9×2+6 1926 = . 58743 (5 + 8 × 7) × 4 × 3 1953 1+9×5+3 = . 26784 2 × (6 + 78) × 4
1286 1×2+8+6 = . 73945 (7 + 39) × 4 × 5
•
1 × 3 × (4 + 8) 1348 = . 56279 (5 + 6 × 27) × 9
•
1953 1 × 9 + 53 = . 84672 8×4×6×7×2
•
2871 2 × 87 × 1 = . 43956 (439 + 5) × 6
•
(1 + 3 × 6) × 8 1368 = . 59472 (5 + 9) × 472
•
21 + 63 2163 = . 84975 (8 + 4 × 9) × 75
•
2 + 8 × (7 + 4) 2874 = . 53169 5 × (31 + 6) × 9
957 9+5+7 = . 61248 (6 + 1) × 24 × 8
•
(1 + 4) × 3 × 5 1435 = . 26978 2 × (697 + 8)
•
2174 2 + 174 = . 36958 (369 + 5) × 8
•
2915 (2 + 9) × 15 = . 34768 (34 + 7) × 6 × 8
962 (9 + 6) × 2 = . 15873 (158 + 7) × 3
•
1438 (1 + 4) × 3 × 8 = . 95627 95 × 6 × 2 × 7
•
2178 2×1×7+8 = . 43659 (4 + (3 + 6) × 5) × 9
•
2916 2+9+1+6 = . 35478 3 + (5 × 4 + 7) × 8
9 × (6 + 2) 962 = . • 84175 84 × 1 × 75
•
1+4+5+8 1458 = . 62937 (6 × 2 × 9 + 3) × 7
•
2 + 18 + 7 2187 = . 35964 (3 × 5 + 96) × 4
•
2943 (2 + 9 + 4) × 3 = . 76518 (7 + 6) × 5 × 18
964 (9 + 6) × 4 • = . 78325 (7 + 8) × 325
1458 1 × (4 + 5) × 8 = . 73629 7 + 3629
2349 (2 + 3) × 4 + 9 • = . 75816 (75 + 81) × 6
•
•
3276 32 × (7 + 6) = . 15498 (1 + 5 × 49) × 8
•
3276 3×2×7×6 = . 18954 18 × 9 × (5 + 4)
•
3456 (3 + 45) × 6 = . 98172 9 + 8172
•
3465 (3 + 46) × 5 = . 21978 2 × (1 + 97 × 8)
•
3485 (3 + 48) × 5 = . 19762 19 × 76 + 2
951 9+5×1 = . 68472 6 × (8 + 4) × 7 × 2
•
•
9 × (5 + 2) 952 = . 34816 3 × 48 × 16
•
•
956 (9 + 5) × 6 = . 37284 (37 + 2) × 84
•
•
•
•
9+6+5 965 = . 12738 (12 + 7 × 3) × 8
1593 1 × 59 × 3 • = . 27864 (2 + 7) × 86 × 4
965 9+6+5 = . 28371 28 × 3 × 7 × 1
•
21 2+1 = . 398475 (3 + 9 × 84) × 75
•
•
26 2+6 = . 539487 53 × 9 × 4 × 87
•
•
•
•
1635 1×6×3×5 = . 98427 9 × (84 + 2) × 7
•
•
1953 1 × 9 + 53 = . 78624 78 × (6 + 2) × 4
2349 (2 + 3) × 4 + 9 = . 86751 (8 + 6 + 7) × 51 2354 (2 × 3 + 5) × 4 = . 69871 (6 + 9) × 87 + 1
1672 1×6×7×2 = . 35948 3 × (594 + 8)
•
•
1725 (1 + 7 × 2) × 5 = . 39468 3 × (94 × 6 + 8)
•
2394 2×3+9+4 = . 15876 (1 + 5 + 8 + 7) × 6
•
3492 (3 + 4 + 9) × 2 = . 57618 (5 × (7 + 6) + 1) × 8
31 3×1 = . 784269 (7 + 8426) × 9
•
1738 1×7+3+8 = . 29546 (2 + 9 × 5 + 4) × 6
•
2457 2 × (4 + 5) × 7 = . 13689 13 + 689
•
3549 3 × (5 + 4) × 9 = . 81627 81 × (62 + 7)
31 3+1 = . 796824 7 × 9 × 68 × 24
•
1 × 74 + 6 1746 = . 82935 8 × (2 + 93) × 5
•
2 × 4 + 61 2461 = . 37985 3 × (7 × 9 + 8) × 5
•
3 × (5 + 6) × 4 3564 = . 18927 1 + (8 + 92) × 7
2375 2 × 375 = . 14896 (1 + 48) × 96
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
3618 3 × (6 + 1 × 8) = . 29547 (29 + 5 × 4) × 7
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4392 43 + 9 × 2 = . 18576 1 × (8 + 5 × 7) × 6
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5184 (5 + 1) × 8 × 4 = . 32967 3 + 29 × 6 × 7
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6174 6+1+7+4 = . 89523 ((8 + 9) × 5 + 2) × 3
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3645 (3 + 6 × 4) × 5 = . 18927 1 + (8 + 92) × 7
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4523 (4 + 5 × 2) × 3 = . 76891 7 × 6 × (8 + 9 × 1)
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5238 (5 × 2 + 3) × 8 = . 49761 (4 + 9) × 76 × 1
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6179 6 × 17 + 9 = . 25384 (2 × 53 + 8) × 4
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6192 (6 + 1 + 9) × 2 = . 48375 (4 × 8 + 3) × 7 + 5
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6214 (6 × 2 + 1) × 4 = . 38957 3+8+9×5×7
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6237 6×2+3×7 = . 48195 (4 × 8 + 19) × 5
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6278 6 + 2 + 78 = . 19345 (19 + 34) × 5
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6318 6 × (31 + 8) = . 27945 (2 × 7 + 9) × 45
3645 (3 + 6) × 45 • = . 91728 91 × 7 × 2 × 8
4536 (4 + 5 + 3) × 6 • = . 98721 9 × 87 × 2 + 1
5 + 328 5328 = . • 14976 (149 + 7) × 6
3+6+5×4 3654 = . • 91728 (9 + 17) × 28
4563 (4 × 5 + 6) × 3 • = . 72891 7 × 2 × 89 × 1
5328 (5 + 3) × 28 • = . 19647 (19 × 6 + 4) × 7
4563 (4 × 5 + 6) × 3 = . 91728 (91 + 7) × 2 × 8
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4617 46 + 1 + 7 = . 82935 (8 + 2 × 93) × 5
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5376 (5 + 3) × 7 × 6 = . 14928 1 + 4 + 928
3×7×1×5 3715 = . 69842 6 + 984 × 2
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4628 4 × 6 + 28 = . 13795 (1 + 3 × 7 + 9) × 5
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(5 + 4 + 7) × 2 5472 = . 19836 1 × 98 + 3 × 6
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6×3×1+8 6318 = . 49572 (4 + (9 + 5) × 7) × 2
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3725 (3 + 7) × 2 + 5 = . 61984 (6 + 1 × 98) × 4
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4 × 6 + 31 4631 = . 59782 5 × (9 × 7 + 8) × 2
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5472 5 × (4 + 72) = . 39168 (39 + 1) × 68
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6345 (6 × 3 + 4) × 5 = . 29187 2+9×1×8×7
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3726 (3 × 7 + 2) × 6 = . 15984 (1 + 5) × 98 + 4
4671 4 + 67 + 1 • = . 52938 (5 + 29) × 3 × 8
54 + 8 + 1 5481 = . • 39672 3 × (9 + 67) × 2
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6384 (6 + 3) × 8 + 4 = . 15792 (15 + 79) × 2
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6435 (6 + 4 + 3) × 5 = . 81972 819 + 7 + 2
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(6 + 4 + 3) × 5 6435 = . 87219 872 + 1 × 9
5691 5×6×9+1 = . 24738 (24 + 7) × 38
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6498 (6 + 4 + 9) × 8 = . 15732 1 + 5 × 73 + 2
3 + 6 + 54 3654 = . 91872 9 × (1 + 87) × 2
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3672 (3 + 6 + 7) × 2 = . 48195 (4 + 8 × (1 + 9)) × 5
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3762 3 + (7 + 6 × 2) • = . 41895 ((4 + 1) × 8 + 9) × 5
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3821 3×8×2+1 = . 64957 (6 × 4 + 95) × 7
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3 × (8 + 2) × 5 3825 = . 64719 6 × 47 × 1 × 9
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3861 (3 + 8) × 61 = . 25974 (2 + 59) × 74
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3864 3 × (8 + 6) + 4 = . 15792 (15 + 79) × 2
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39 + 27 3927 = . • 18564 1 × (8 + 5) × 6 × 4
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3948 (3 + 9) × 4 + 8 = . 17625 ((1 + 7) × 6 + 2) × 5
(46 + 9) × 7 4697 = . 23851 23 × 85 × 1
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5328 (5 + 32) × 8 = . 41976 4 × (1 + 97 × 6)
5643 (5 + 6 + 4) × 3 = . 97812 (9 + 7 × 8) × 12
(4 + 7 + 3) × 2 4732 = . 81965 8 × (1 + 9) × 6 + 5
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4752 47 + 52 = . 39168 (3 + 9 × 1) × 68
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5712 (5 + 7) × 12 = . 34986 3 × (4 × 9 × 8 + 6)
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6 × (49 + 8) 6498 = . 21375 (2 + 13) × 75
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4769 47 + 6 + 9 = . 38152 (3 × 81 + 5) × 2
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(5 + 7 × 4) × 2 5742 = . 86913 86 + 913
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6512 (6 + 5 × 1) × 2 = . 97384 9 + (7 + 3) × 8 × 4
4851 48 + 51 • = . 37926 (3 + 7 × 9 × 2) × 6
5792 5 × (7 + 9) × 2 • = . 18643 1 × 8 × 64 + 3
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(6 + 5) × 3 × 4 6534 = . 27819 2 + 7 × 8 × (1 + 9)
4 × 9 + 21 4921 = . • 83657 (8 + 3 + 6) × 57
(5 + 7 + 9) × 6 5796 = . • 13248 1 × 3 × 2 × 48
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6534 (6 + 5) × 34 = . 27918 2 × (791 + 8)
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6 × (5 + 7 × 9) 6579 = . 12384 1 × 2 × 384
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6579 6 × (5 + 7 × 9) = . 42183 4 × 218 × 3
5892 5+8+9+2 = . 47136 (4 + 7 × (1 + 3)) × 6
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6794 6 × (7 + 9 × 4) = . 13825 (13 + 8) × 25
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4968 4 + 96 + 8 = . 15732 1 × 57 × 3 × 2
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5841 58 + 41 = . 29736 (2 + 9 + 73) × 6
3978 3+9+7×8 = . 14625 1 × (4 + 6) × 25
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5124 (5 + 1 × 2) × 4 = . 98637 98 + 63 × 7
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3978 3+9+7×8 = . 52416 (52 + 4) × 16
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5148 (51 + 4) × 8 = . 27963 2 + 796 × 3
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5928 (5 + 9) × 28 = . 67431 6 × 743 + 1
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6931 6×9+3+1 = . 27485 (2 × 7 + 4 × 8) × 5
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(4 + 3 × 2) × 5 4325 = . 71968 (7 + 1 + 96) × 8
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5184 (5 + 1) × 8 × 4 = . 26973 26 + 973
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5 × (9 × 7 + 3) 5973 = . 18462 (1 + 84) × 6 × 2
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(69 + 7) × 3 6973 = . 52481 52 × (4 × 8 + 1)
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Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
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6975 6+9+7×5 = . 12834 (1 + 2 × (8 + 3)) × 4
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8352 (8 × 3 + 5) × 2 = . 14976 (1 + 4 + 9) × 7 + 6
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9152 (91 + 5) × 2 = . 38467 3 + (8 + 4) × 67
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6975 6+9+7×5 = . 31248 (3 + 1 + 24) × 8
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8352 (8 + 3 + 5) × 2 = . 46719 46 + 7 × 19
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7×1×2+3 7123 = . 98465 (9 + 8 × 4 + 6) × 5
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8365 8×3+6+5 = . 49712 (4 + 9) × (7 + 1) × 2
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7128 (7 + 1 × 2) × 8 • = . 65934 659 + 3 + 4 •
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7194 7+1+9×4 = . 53628 (5 + 3 × 6 × 2) × 8 7254 7+2+5+4 = . 68913 (6 × 8 + 9 × 1) × 3
8 + (3 + 7) × 9 8379 = . • 21546 2 × (1 + 5 × 4) × 6
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7259 7 × (25 + 9) = . 16348 1 × (63 + 4) × 8
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7346 7+3+4×6 = . 91825 (9 + 1 × 8) × 25
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8379 8 + (3 + 7) × 9 = . 64125 6 × (4 + 1) × 25 (8 × 4 + 1) × 5 8415 = . 39627 (3 + 9 × 6 × 2) × 7
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156 (1 + 5) × 6 = . 978432 9 × 784 × 32
9168 9×1×6×8 = . 37245 (37 + 2) × 45
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162 (1 + 6) × 2 = . 394875 3 × (9 + 4) × 875
9213 9 + 213 = . 68475 6 × (8 + 47) × 5
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16 + 2 162 = . 539487 53 × (9 + 4) × 87
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213 2+1+3 = . 748695 74 × (8 × 6 + 9) × 5
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2 × 16 216 = . 394875 (3 + 9) × 4875
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243 24 + 3 = . 196875 (19 + 6) × 875
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249 2+4+9 = . 157368 15 × (73 + 6) × 8
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2×5+6 256 = . 137984 (1 + 3 × 7) × 98 × 4
9243 92 + 4 × 3 • = . 81765 (8 + 176) × 5
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9268 9 × 2 × (6 + 8) = . 41375 (4 + 1) × 3 × 75 9315 9 × 3 × 15 = . 67482 6 × (7 + 482)
8415 8 × (4 + 1) + 5 = . 72369 (7 + 2 × 3 × 6) × 9
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8415 (84 + 1) × 5 = . 76923 7 × (6 × 92 + 3)
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9387 93 + 8 × 7 = . 15624 (1 + 5 × 6) × 2 × 4
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2×5+8 258 = . 317469 (317 + 4) × 69
(7 + 6 + 3) × 2 7632 = . 59148 (59 + 1) × 4 + 8
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8492 (8 + 4) × (9 + 2) = . 17563 (17 × 5 + 6) × 3
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93 + 8 × 7 9387 = . 21546 (2 + 1 + 54) × 6
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2×6×1 261 = . 348957 3 × (4 + 8 × 95) × 7
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7659 (7 + 6) × 5 + 9 = . 12834 1 × (28 + 3) × 4
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8694 (8 + 6 + 9) × 4 = . 25137 (25 + 13) × 7
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9432 (9 + 4 + 3) × 2 = . 17685 1×7+6×8+5
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261 2×6×1 = . 489375 4 × (8 × 9 + 3) × 75
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7 × (8 + 1) × 2 7812 = . 34596 3 × (4 × 5 × 9 + 6)
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8694 (8 + 6 + 9) × 4 = . 35721 357 + 21
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9612 9×6×1×2 = . 58473 584 + 73
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(2 + 6) × 3 263 = . 174895 (17 + 4) × 8 × 95
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7935 (7 + 9) × 3 × 5 = . 68241 6 × 8 × (2 + 41)
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8694 (8 + 6 + 9) × 4 = . 52731 527 + 31
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9614 (9 × 6 + 1) × 4 = . 25783 2 × 5 × (7 × 8 + 3)
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264 2×6+4 = . 193875 (1 + 9 × 3 × 87) × 5
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7952 7 × (9 + 5 + 2) = . 43168 4 × (3 + 16) × 8
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8712 87 + 12 = . 45936 (4 × 5 + 9) × 3 × 6
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9 × (6 + 1 × 8) 9618 = . 35724 (3 + 57 × 2) × 4
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2 × (8 + 4) 284 = . 613795 6 × 13 × 7 × 95
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291 2×9×1 = . 376845 37 × (6 + 8) × 45
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29 + 7 297 = . 163548 (1 + 6) × 354 × 8
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314 3+1×4 = . 725968 7 × (25 + 9) × 68
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315 3×1×5 = . 469728 4 × (697 + 2) × 8
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315 (3 + 1) × 5 = . 486927 4 + 8 × 6 × 92 × 7
9856 98 + 5 × 6 = . 31724 (31 + 72) × 4
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315 (3 + 1) × 5 = . 649278 64 × 92 × 7 + 8
12 + 6 126 = . 398475 (3 + 9 × 84) × 75
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315 3×1×5 = . 689472 6 × 8 × 9 × (4 + 72)
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9315 9 × (3 + 1) × 5 = . 72864 (7 × 2 + 8) × 64
8136 8 × 1 × (3 + 6) • = . 92547 (9 + 2 × 54) × 7
8712 (8 + 7 + 1) × 2 • = . 93654 9 × 36 + 5 × 4
96 + 48 9648 = . • 27135 27 × 1 × 3 × 5
8176 8×1×7×6 = . 92345 (9 + 2) × 345
8715 8×7×1×5 • = . 26394 2 + (6 + 3) × 94
9683 (9 + 6 + 8) × 3 • = . 47152 4 × 7 × (1 + 5) × 2
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8246 8 × 2 + 46 = . 79135 (7 + 9 + 1) × 35
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8316 (8 + 3) × 16 = . 45927 45 + 927
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(8 + 3) × 25 8325 = . 17649 1 × 7 + 64 × 9
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8 + 32 + 6 8326 = . 51947 (5 + 1 × 9 × 4) × 7
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8729 (8 × 7 + 2) × 9 = . 34615 3 × 46 × 15
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8764 (8 + 76) × 4 = . 39125 (3 + 9) × 125
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8975 (8 + 97) × 5 = . 13642 (1 + 3 × 6) × 42
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9 × 1 × 26 9126 = . 43875 (4 + 3 + 8) × 75
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9 + 76 + 5 9765 = . 21483 2 × (1 + 4 × 8) × 3 9765 9 + 76 + 5 = . 31248 3 × 1 × 2 × 48
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
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324 3 + 24 417 4 + 17 = .• = . 185976 (1 + 8 × 5) × 9 × 7 × 6 293568 (2 + 9) × 3 × 56 × 8
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612 6×1×2 = . 439875 (43 + 9 × 8) × 75
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819 8+1+9 = . 563472 (56 × 3 + 4) × 72
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324 3 × 24 = . 197568 (1 + 97) × 56 × 8
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426 4+2×6 = . 931875 (9 + 31) × 875
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612 6×1×2 = . 748935 (7 + 4) × 89 × 3 × 5
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823 8+2+3 = . 176945 (1 + 7 × 6) × (9 + 4) × 5
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324 (3 + 2) × 4 = . 597861 (597 + 8) × 61
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427 (4 + 2) × 7 = . 139568 (1 + 3 × 95) × 6 × 8
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6+1×2 612 = . 753984 7 × (5 + 39) × 8 × 4
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837 8×3+7 = . 126495 (1 + 26 × 4 × 9) × 5
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3 × (2 + 4) 324 = . 791568 (7 + 915 × 6) × 8
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612 6+1+2 432 4 + 32 = . = .• 9 × (375 + 8) × 4 185976 (1 + 8 × 5) × 9 × 7 × 6 937584
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8×3+7 837 = . 469125 (4 + 691) × 25
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324 3+2+4 = . 795168 (7 + 9 × 51 × 6) × 8
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432 4×3×2 = . 791568 (7 + 915 × 6) × 8
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6×1×2 612 = . 948753 9 × (4 × 8 + 7) × 53
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846 8+4+6 = . 153972 (1 + 5) × 39 × 7 × 2
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3×2+7 327 = . 416598 (4 + 165) × 98
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435 4×3×5 = . 279618 (2 + 79 × 61) × 8
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618 6+1×8 = . 749325 7 × (4 + 93) × 25
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846 8+4+6 = . 391275 (3 + 9 × 12) × 75
328 3×2×8 = . 145796 (1 + 45 × 79) × 6
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468 4+6×8 = . 192375 1 × 9 × 2375
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864 8+6×4 = . 192375 1 × (92 + 3) × 75
468 4+6×8 342 3 × (4 + 2) = . • 739125 = 73 × 9 × 125 . 178695 (1 + 7 × (8 + 6)) × 95
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873 8+7+3 = . 624195 (62 + 4) × 195
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879 8+7+9 = . 136245 1 × 3 × 62 × 4 × 5
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912 (9 + 1) × 2 = . 576384 5 × (76 + 3) × 8 × 4
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9+1×8 918 = . 365472 (3 × 6 × 5 + 4) × 72
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927 9+2+7 = . 183546 1 × (8 + 3) × 54 × 6
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483 4+8×3 = . 219765 2 × (1 + 97) × 65
621 6×2×1 = . 395784 (3 + 95) × 78 + 4 621 6 × (2 + 1) = . 589743 (5 + 8 × 9) × 74 × 3 6×2+4 624 = . 173589 1 + (7 + 3) × 5 × 89
(3 + 4) × 2 342 = . 785916 7 × (85 × 9 + 1) × 6
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342 (3 + 4) × 2 = . 915876 91 × (58 × 7 + 6)
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342 34 + 2 = . 961875 9 × 6 × 1875
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3×4+7 347 = . 168295 (1 + 6 × 8 × 2) × 95
513 5×1×3 681 6×8×1 • = . 289674 (2 + 8 × 96) × (7 + 4) • 493725 = 4 × (9 + 3) × 725 .
486 48 + 6 = . 739125 73 × 9 × 125 489 4+8+9 = . 125673 (1 + 256) × 7 × 3
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624 6 × 24 = . 189735 (1 + 8) × 973 × 5 6+3+9 639 = . 158472 (1 × 58 + 4) × 72
513 5 + 13 6 + 8 × (3 + 7) 6837 351 3+5+1 • = . • 689472 = 6 × 8 × 9 × 4 × 7 × 2 . • 29415 = 2 × (9 × 4 + 1) × 5 . 248976 2 × (4 + 8 × 9) × 7 × 6 351 3×5×1 • = . 467298 4 + 67 × 298
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531 5+3×1 6×8+4 = . 684 269748 ((2 + 6) × 9 × 7 + 4) × 8• 192375 = (192 + 3) × 75 .
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927 9+2×7 = . 681345 (6 × 8 + 1) × 345
351 3×5+1 • = . 928746 9 × 28 × 7 × 4 × 6
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543 5+4+3 = . 269871 (2 × 6 + 9 × 8) × 71
684 (6 + 8) × 4 = . • 312759 31 × 2 × 7 × 59
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938 9+3+8 = . 142576 1 × 4 × 2 × 5 × 76
357 (3 + 5) × 7 = . 196248 1 × 962 × 4 × 8
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543 5 + 43 = . 691782 6 × 91 × 7 × 8 × 2
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693 6+9×3 = . 125874 (1 + 2 × 5 × 8) × 74
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945 (9 + 4) × 5 = . 138726 13 × (8 + 726)
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378 (3 + 7) × 8 = . 156492 15 × 6 × 4 × 92
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546 5×4+6 = . 137928 (13 × 7 × 9 + 2) × 8
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693 6+9+3 = . 724185 (7 + 2) × 418 × 5
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(9 + 4) × 5 945 = . 273861 273 × (8 + 61)
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394 3+9+4 = . 217685 (2 + 17 × 6) × 85
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549 5+4+9 = . 127368 (12 × 7 + 3) × 6 × 8
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743 7+4+3 = . 962185 (96 + 2) × 185
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957 9 × (5 + 7) = . 214368 21 × 4 × 36 × 8
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396 (3 + 9) × 6 = . 145728 (1 + 45) × 72 × 8
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549 5+4+9 = . 237168 (23 × 7 + 1) × 6 × 8
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759 7×5+9 = . 183264 1 × 83 × 2 × 64
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972 9+7+2 = . 143856 (1 + 438 + 5) × 6
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396 (3 + 9) × 6 = . 571824 57 × 1824
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583 (5 + 8) × 3 = . 429671 429 × 67 × 1
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783 7+8+3 972 9 + 72 = . • = . 129456 (1 + (2 + 9) × 45) × 6 183456 (1 + 8 × 34) × 56
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412 4+1×2 = . 837596 (8 + 3 × 75 × 9) × 6
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597 5+9+7 = . 162384 1 × 6 × 238 × 4
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813 8×1×3 = . 247965 24 × (7 + 9 × 6) × 5
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987 98 + 7 = . 461352 4 × 6135 × 2
Inder J. Taneja
2.2
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
Without Zero and Non Extendable to Multiple of Ten
This subsection bring selfie fractions with addition and multiplication without zero and are not extendable to multiple o 10 as subsection 2.1. Since the work is with out zero and is with different digits, obviously, the results are maximum up to nine digits.
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41 4+1 = . 369 3 × (6 + 9)
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26 + 1 261 = . 348 3 × (4 + 8)
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576 5+7+6 = . 832 8×3+2
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247 2×4+7 = . 5681 5 × (68 + 1)
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43 4+3 = . 258 2+5×8
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276 2×7+6 = . 483 4×8+3
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5+9+4 594 = . 627 6×2+7
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342 3 × (4 + 2) = . 1786 1 + 7 + 86
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51 5+1 = . 289 2 × (8 + 9)
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285 2+8+5 = . 361 3×6+1
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639 6 × (3 + 9) = . 781 7 + 81
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342 3 × (4 + 2) = . 1957 1 + 95 + 7
71 7+1 • = . 639 6 × (3 + 9)
287 28 + 7 • = . 369 3 × (6 + 9)
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7+5+6 756 = . 924 9×2+4
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348 3+4+8 = . 1276 1 + (2 + 7) × 6
136 1×3+6 • = . 952 9 × (5 + 2)
289 2 × (8 + 9) • = . 357 35 + 7
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81 8+1 = . 3429 3 + 42 × 9
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357 3+5+7 = . 2618 2 + 6 × 18
289 2 × (8 + 9) • = . 561 5 + 61
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128 1+2×8 = . 3456 3 + 456
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153 15 + 3 • = . 289 2 × (8 + 9)
3+7+2 372 = . 6541 6 + 5 × 41
132 1+3+2 • = . 8976 8 × (9 + 7 × 6)
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378 3+7+8 = . 1596 1 + 5 × (9 + 6)
1+3+4 134 = . • 5762 57 × 6 + 2
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378 3+7+8 = . 5691 5×6×9+1
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412 4 × (1 + 2) = . 5768 (5 + 7) × (6 + 8)
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413 4+1×3 = . 5782 (5 + 7) × 8 + 2
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423 42 + 3 = . 7896 7 × 8 × (9 + 6)
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437 4×3+7 = . 2691 26 + 91
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4+5+9 459 = . 1683 1 × 6 × (8 + 3)
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463 4+6×3 = . 1852 1 + 85 + 2
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473 4+7×3 = . 1892 1 × 8 + 92
213 2 × (1 + 3) = . 8946 8 × (9 × 4 + 6)
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526 5 × (2 + 6) = . 9731 9 + 731
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163 1+6×3 = . 489 48 + 9 164 1 × (6 + 4) = . 738 7 + 38
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168 1×6+8 = . 324 3 + 24
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1×6+8 168 = . 432 4 + 32
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168 1×6+8 = . 972 9 + 72
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194 1+9+4 = . 582 5×8+2
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195 1+9+5 = . 286 2×8+6
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2+1×6 216 = . 837 8×3+7
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216 2+1+6 = . 984 9+8×4
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3+2×9 329 = . 846 8 + 46 369 3 × (6 + 9) = . 451 4 + 51 374 3 × (7 + 4) = . 918 9 × (1 + 8) 378 3+7+8 = . 546 5×4+6 378 3+7+8 = . 651 6×5+1
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137 1×3+7 = . 9864 9 × 8 × (6 + 4) 142 1×4+2 = . 6958 6 × (9 + 5 × 8) 152 1+5+2 = . 6498 6 × (49 + 8) 1+7+2 172 = . 3698 3 × 69 + 8
385 (3 + 8) × 5 = . 462 4 + 62
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396 3 × (9 + 6) = . 528 52 + 8
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4×3+7 437 = . 529 5+2×9
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437 4×3+7 = . 621 6 + 21
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231 2×3+1 = . 594 5+9+4
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468 4+6×8 = . 729 72 + 9
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234 2+3+4 = . 1976 (1 + 9) × 7 + 6
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5×2+8 528 = . 1496 (1 + 4) × 9 + 6
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231 23 + 1 = . 847 8 × (4 + 7)
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513 51 + 3 = . 684 6 × (8 + 4)
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236 2×3+6 = . 1947 1 × 9 × (4 + 7)
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529 5+2×9 = . 1748 17 × 4 + 8
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247 2+4+7 = . 361 3×6+1
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5+2 52 = . 1768 17 × (6 + 8)
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2+3+7 237 = . 1659 1 × 6 × (5 + 9)
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529 5+2×9 = . 1863 18 + 63
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176 1+7+6 = . 5984 59 × 8 + 4 213 2+1×3 = . 5964 (5 + 9) × (6 + 4)
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
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531 5+3+1 = . 4897 4+8×9+7
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632 6+3×2 = . 7584 (7 + 5) × (8 + 4)
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867 8 + 67 = . 5491 5 × (4 + 91)
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1254 1×2+5×4 = . 7638 7×6×3+8
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531 5+3×1 = . 8496 8 × 4 + 96
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638 6+3+8 = . 5742 5 + 74 × 2
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873 8+7+3 = . 1649 1+6×4+9
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1254 1 + 2 + 54 = . 8976 8 × (9 + 7 × 6)
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537 5+3+7 = . 2864 2 × 8 + 64
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651 6×5+1 = . 4872 4 × (8 × 7 + 2)
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891 8+9+1 = . 2376 2×3×7+6
•
1258 12 × 5 + 8 = . 4736 4 + 7 × 36
•
537 5+3+7 = . 4296 4 × 2 × (9 + 6)
•
891 8+9+1 = . 2475 2 + 4 × (7 + 5)
•
1386 1 × 38 + 6 = . 2457 2 × (4 + 5 × 7)
•
541 5+4×1 = . 9738 9×7+3+8
•
1 × 3 × (8 + 6) 1386 = . 2574 2 × (5 × 7 + 4)
•
56 + 7 567 = . 3429 3 + 42 × 9
•
1428 14 × (2 + 8) = . 5967 5 × 9 × (6 + 7)
•
1+4×2+8 1428 = . 9576 9 × (5 + 7) + 6
•
1436 1 × 4 × (3 + 6) = . 8257 8 × 25 + 7
•
1452 1×4+5×2 = . 7986 7×9+8+6
•
1472 1 + (4 + 7) × 2 = . 9856 98 + 56
•
1485 1+4+8×5 = . 2673 2 + 6 + 73
•
1536 1+5+3×6 = . 4928 49 + 28
•
1+5+4+7 1547 = . 2639 2+6×3+9
9+5+4 954 = . 3816 3 × (8 + 16)
•
1548 1 × 5 × (4 + 8) = . 2967 2×9×6+7
•
954 9+5+4 = . 6731 6×7×3+1
•
1584 (1 + 5) × (8 + 4) = . 7326 7 + 326
•
956 9+5+6 = . 3824 38 × 2 + 4
•
1596 1 + 5 × (9 + 6) = . 4872 4 × (8 × 7 + 2)
792 7+9+2 = . 5148 5 + 14 × 8
•
972 9+7+2 = . 4536 4 × (5 × 3 + 6)
•
1672 1×6×7+2 = . 3458 (3 + 4) × (5 + 8)
798 7+9+8 = . 4256 4 × (2 + 5 × 6)
•
984 9+8×4 = . 2376 23 + 76
•
1679 1+6+7+9 = . 3285 3 × (2 + 8 + 5)
•
594 5+9+4 = . 1386 1 × 3 × (8 + 6)
594 5+9+4 • = . 1683 1×6×8+3
652 6+5×2 • = . 1793 17 + 9 × 3
•
•
•
•
5+9+4 594 = . • 2178 2 + (1 + 7) × 8 • •
594 5+9+4 = . 2871 2 × 8 + 71 594 5+9+4 = . 3267 3 × (26 + 7) 596 5+9+6 = . 2384 2 × 38 + 4
•
614 6+1×4 = . 7982 (7 + 9) × 8 + 2 618 6+1+8 = . 4532 4 + 53 × 2
621 6+2+1 • = . 3795 3+7+9×5
726 7 + 26 = . 1584 (1 + 5) × (8 + 4) 72 + 6 726 = . 4598 4 + 5 × 98
918 9+1×8 • = . 3456 34 + 5 × 6 9+1×8 918 = . • 4536 4 × (5 × 3 + 6)
735 7+3+5 = . 8624 86 × 2 + 4 756 7+5+6 = . 1932 1 + 9 × (3 + 2) •
• •
684 6 × (8 + 4) = . 1539 153 + 9
759 7 + 59 = . 1863 18 × (6 + 3) •
784 7 × (8 + 4) • = . 2156 21 × (5 + 6)
•
792 7+9+2 = . 1364 1 + 3 × (6 + 4)
912 9+1+2 = . 3876 3 × (8 + 7) + 6
9+1+5 915 = . • 2684 2×6+8×4
• •
•
682 6+8×2 = . 1953 1 + 9 + 53
•
• •
•
679 6+7+9 = . 1358 1+3+5×8
8+9+1 891 = . • 4653 4+6×5×3
918 9+1×8 = . 5346 53 + 46 9×2+8 928 = . 4176 41 + 76 931 9×3+1 = . 4256 4 × (2 + 5 × 6)
627 (6 × 2) + 7 = . • 1485 1+4+8×5
•
627 6 + 27 • = . 1843 1+8×4×3
•
•
627 6×2+7 = . 3498 34 + 9 × 8
•
819 8 + 19 = . 4732 4 × (7 + 32)
•
24 2+4 = . 18796 1 + 87 × 9 × 6
•
1683 1×6×8+3 = . 2574 2 × (5 × 7 + 4)
•
627 6×2+7 = . 3894 3 × 8 + 94
•
861 86 + 1 = . 4592 4 + 5 × 92
•
81 8+1 = . 45639 4 + 563 × 9
•
1683 (1 + 6) × (8 + 3) = . 2754 2 × 7 × (5 + 4)
•
627 6×2+7 = . 9834 98 × 3 + 4
•
864 8+6+4 = . 1392 1×3×9+2
•
1236 1 × 2 × (3 + 6) = . 5974 59 + 7 × 4
•
1683 1×6+8+3 = . 4257 4×2+5×7
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
1683 16 + 83 = . 7259 7 × (2 + 59)
•
2178 2×1×7+8 = . 5346 5 + 3 + 46
•
2871 2 × 8 + 71 = . 4356 4 × 3 × (5 + 6)
•
3854 3+8×5+4 = . 7216 72 + 16
•
1694 1×6+9×4 = . 3872 3 × 8 + 72
•
2184 2 × (1 + 8 + 4) = . 9576 9 × (5 + 7) + 6
•
2871 2+8×7×1 = . 4653 4+6×5×3
•
3912 (3 + 9) × 12 = . 8476 8 + 4 × 76
•
1+7×3×4 1734 = . 6528 6 × 52 + 8
•
2 + 18 + 7 2187 = . 4536 4 × (5 + 3 + 6)
•
3 × (12 + 9) 3129 = . 4768 4 × 7 + 68
•
3 × (91 + 4) 3914 = . 5768 5 × (76 + 8)
•
1 + 7 + 46 1746 = . 3298 3 × 2 × (9 + 8)
•
2 + 18 + 7 2187 = . 5346 5×3×4+6
•
3154 3 + 1 × 54 = . 6972 6 × 9 + 72
•
3 × (9 + 1 + 5) 3915 = . 7482 7 × (4 + 8) + 2
•
1748 1×7+4+8 = . 6532 65 + 3 × 2
•
2196 2 + (1 + 9 + 6) = . 5734 5×7+3×4
•
(3 + 1) × 6 + 8 3168 = . 4257 4×2+5×7
•
3982 3+9+8+2 = . 6154 6×1×5+4
•
17 + 6 × 8 1768 = . 4352 4 + 3 × 52
•
2197 2 × (19 + 7) = . 4563 45 + 63
•
3192 3 × (1 + 9 × 2) = . 5768 5 + 7 × (6 + 8)
•
4+1+7+2 4172 = . 9536 9+5+3×6
•
1782 17 + 8 × 2 = . 3456 34 + 5 × 6
•
2 + 4 + 13 2413 = . 6985 6+9+8×5
•
3248 (3 + 2) × 4 + 8 = . 5916 5×9×1+6
•
4182 41 × (8 + 2) = . 5967 5 × 9 × (6 + 7)
•
1782 1+7+8+2 = . 3465 3 × (4 + 6) + 5
•
2431 2×4+3×1 = . 5967 5+9+6+7
•
3+2×6+7 3267 = . 4158 4×1×5+8
•
4321 43 × 2 + 1 = . 9685 (9 + 6) × (8 + 5)
•
1782 17 + 8 + 2 = . 4356 4×3×5+6
•
2 + 4 × (9 + 7) 2497 = . 6583 6 × (5 + 8 × 3)
•
3 × 2 × (7 + 6) 3276 = . 4158 41 + 58
•
4325 4 + 3 × (2 + 5) = . 9861 9+8×6×1
•
1782 17 + 8 × 2 = . 4536 4 × (5 × 3 + 6)
•
2514 2 × (5 + 1 × 4) = . 9637 9 + 6 × (3 + 7)
•
3286 3 + 2 × (8 + 6) = . 5194 5×1×9+4
•
4326 (4 + 3) × (2 + 6) = . 5871 5 × (8 + 7) + 1
•
1782 1+7+8+2 = . 4653 4 × (6 + 5) + 3
•
2 × (5 + 3) + 8 2538 = . 7614 7 + 61 + 4
•
3297 3+2+9+7 = . 8164 8×1×6+4
•
4536 4+5+3+6 = . 7812 7 + 8 × (1 + 2)
•
1×8×9+6 1896 = . 5372 5 + 3 × 72
•
2573 2×5+7×3 = . 8964 8 + 96 + 4
•
3416 34 + 1 × 6 = . 5978 5+9+7×8
•
4536 45 × (3 + 6) = . 8792 87 × 9 + 2
•
1936 1×9×3+6 = . 4752 4 + 75 + 2
•
2 × (5 + 7 × 4) 2574 = . 6318 6 × 3 × (1 + 8)
•
3584 3 + 5 × (8 + 4) = . 9216 9 × (2 + 16)
•
4683 4+6+8+3 = . 5129 5+1×2×9
•
1938 1 + 93 + 8 = . 6574 6 × 57 + 4
•
2596 2 × (5 + 9 × 6) = . 3784 3×7×8+4
•
3596 3+5+9×6 = . 4872 4 + 8 + 72
•
4689 4+6+8+9 = . 5731 5 + 7 × (3 + 1)
•
1952 1 × 9 + 52 = . 3648 3 × (6 + 4 × 8)
•
2694 2+6×9+4 = . 5837 (5 + 8) × (3 + 7)
•
3627 3 × (6 + 2) + 7 = . 8541 8 × (5 + 4) + 1
•
4 + 7 × 16 4716 = . 8253 8 × 25 + 3
•
1985 1+9+8×5 = . 4367 43 + 67
•
2714 2 + 7 + 14 = . 3658 3×6+5+8
•
3762 3 × (7 + 6 × 2) = . 4158 4 + 1 + 58
•
4756 4+7+5×6 = . 8932 8×9+3+2
•
2168 2 × 16 + 8 = . 3794 3+7×9+4
•
2716 2×7×1×6 = . 8439 84 × 3 + 9
•
3762 3 × (7 + 6 × 2) = . 5148 5 × 14 + 8
•
4832 4 × 8 + 32 = . 6795 6 + 79 + 5
•
2176 2 × (1 + 7 + 6) = . 5984 5×9+8×4
•
2756 2+7×5×6 = . 9841 9 × 84 + 1
•
3796 3×7+9+6 = . 8541 8 × 5 + 41
•
4851 4 × (8 + 5 + 1) = . 6237 6 × (2 + 3 + 7)
•
2178 2×1×7+8 = . 3465 3 × (4 + 6) + 5
•
2761 2+7×6×1 = . 8534 8 × (5 + 3 × 4)
•
3 × (8 + 16) 3816 = . 5247 5 + 2 × 47
•
4856 4+8×5+6 = . 9712 97 + 1 + 2
•
2×1×7+8 2178 = . 4653 4 × (6 + 5) + 3
•
2871 2+8×7×1 = . 3564 3 + 5 + 64
•
3852 3 × (8 + 5 + 2) = . 9416 94 + 16
•
4872 4 × (8 × 7 + 2) = . 5691 5×6×9+1
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
4891 4 × (8 × 9 + 1) = . 5762 57 × 6 + 2
•
178 1+7+8 = . 23496 234 × 9 + 6
•
518 5 × (1 + 8) = . 37296 3 × 72 × (9 + 6)
•
821 8+2+1 = . 45976 4 + (5 + 97) × 6
•
4938 49 + 3 + 8 = . 5761 5 × (7 + 6 + 1)
•
214 2+1×4 = . 76398 7 × 6 × 3 × (9 + 8)
•
531 5 × (3 + 1) = . 28674 2 × (8 × 67 + 4)
•
824 8+2×4 = . 19673 1+9×6×7+3
•
4 + 9 × (8 + 2) 4982 = . 6731 6×7×3+1
•
231 2 × (3 + 1) = . 47586 4 × (7 × 58 + 6)
•
541 5+4+1 = . 63297 6 × 3 × (2 + 9 × 7)
•
824 8 + 24 = . 51397 51 × 39 + 7
•
5×1×9+4 5194 = . 7632 7 + 63 + 2
•
8 + 36 836 = . 19247 1 + 92 × (4 + 7)
•
5481 5×4+8+1 = . 6237 6×2+3×7
•
836 8 + 36 = . 25194 2 × 51 × (9 + 4)
•
837 8+3+7 = . 15624 1 × 56 × (2 + 4)
•
837 8×3+7 = . 24516 2 × 451 + 6
•
8 × (5 + 2) 852 = . 14697 14 × (6 + 9 × 7)
•
•
5497 5 + 4 × (9 + 7) = . 8126 8 × 12 + 6
234 2×3+4 • = . 57681 5 × (7 + 6 × 81)
•
•
56 + 4 + 2 5642 = . 7189 7+1×8×9
•
573 5+7+3 = . 19864 (1 + 9) × (8 × 6 + 4)
•
913 9 + 13 = . 25647 2 + 56 × (4 + 7)
•
312 3 + 12 = . 56784 5 × 6 × (7 + 84)
•
584 5×8+4 = . 36792 36 × 7 × (9 + 2)
•
9+1+6 916 = . 23587 2 × 3 + 58 × 7
•
316 3+1+6 = . 59724 5 × 9 × 7 × (2 + 4)
•
592 5+9+2 = . 63714 6 × (3 + 71 × 4)
•
918 9 + (1 + 8) = . 57324 5 × 7 × 32 + 4
•
321 3 × (2 + 1) = . 45796 4 × (5 × 7 × 9 + 6)
•
596 5+9+6 = . 17284 1 × 72 × 8 + 4
•
9+3×6 936 = . 74152 7 + 41 × 52
324 3 × (2 + 4) • = . 18576 18 × 57 + 6
629 6+2+9 • = . 14578 1 × 4 + 5 × 78
•
954 9+5+4 = . 28673 2 + 8 × 67 + 3
418 4 + 18 • = . 35967 3+5×9×6×7
6+5+4 654 = . • 13298 (1 + 32) × 9 + 8
•
965 9+6+5 = . 24318 2 × (4 + 31 × 8)
428 4 × (2 + 8) • = . 19367 1 + 9 × 3 × 67
• 671 6+7+1 • = . 25498 (2 + 5) × (4 + 9 × 8)
9+7×3 973 = . 85624 8 × 5 × (62 + 4)
•
63 6+3 = . 142597 1 + 42 × 5 × 97
•
1248 1×2×4+8 = . 53976 5 + 3 + 9 × 76
•
1269 1+2+6×9 = . 83754 8 + 3754
5841 5×8+4×1 • = . 6372 6+3×7×2 5916 5 + 9 × (1 + 6) • = . 7482 7 × (4 + 8) + 2
•
•
•
5+9+1×8 5918 = . 7263 7+2+6×3
6714 6 × (7 + 1 + 4) = . 9325 93 + 2 + 5 6842 6+8+4×2 = . 7153 7+1+5×3 7293 7 + 29 + 3 = . 8415 8 × (4 + 1) + 5 7931 7×9+3×1 = . 8652 (8 + 6) × 5 + 2
•
8742 (8 + 7) × 4 + 2 = . 9165 (9 + 1) × 6 + 5
•
•
124 1+2×4 = . 97836 9 × (783 + 6)
•
•
132 1×3+2 = . 57948 5 × (7 + 9 × 48)
•
132 1+3+2 = . 87956 8 + 7 × 95 × 6
•
•
5+6+7 567 = . 24381 2 × 43 × (8 + 1)
291 2 × (9 + 1) = . 45687 4 + 56 × 8 × 7
•
6149 6+1+4×9 = . 8723 8×7+2+3
•
561 5+6×1 = . 47328 4 × (7 × 32 + 8)
•
5742 (5 + 7 × 4) × 2 • = . 8961 (8 + 9) × 6 + 1
•
268 2+6+8 = . 14539 14 × (53 + 9)
•
549 5+4+9 = . 16287 1 × 6 × (2 + 87)
276 2+7+6 = . 13984 1 + 3 + 9 × 84
•
•
243 2+4+3 = . 15768 1 × 576 + 8
•
429 4 + 29 = . 17563 1 + 75 × 6 × 3
•
567 5+6+7 = . 31248 31 × (24 + 8)
6×7+4 674 = . 15839 1+5×8×3×9
452 4+5+2 = . 37968 (3 + 7 × 9) × (6 + 8)
•
472 4+7×2 = . 95816 9 × 58 × (1 + 6)
•
732 7+3+2 = . 65941 6×5×9×4+1
•
1274 1×2+7+4 = . 56938 5 + 6 × (9 + 3) × 8
•
482 4 × (8 + 2) = . 65793 65 × 7 × (9 + 3)
•
742 7 × (4 + 2) = . 93651 93 × (6 + 51)
•
1376 1+3×7+6 = . 29584 2 × 9 + 584
•
491 4+9×1 = . 62357 6 + 235 × 7
•
7 × (6 + 2) 762 = . 89154 8 × 91 × (5 + 4)
•
1392 1×3×9+2 = . 75648 7 × 56 × 4 + 8
732 7+3+2 = . 16958 1×6×9×5+8
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
1456 1+4+5+6 = . 73892 7 × (3 × 8 + 92)
•
1836 1×8+3+6 = . 54972 5+4×9×7×2
•
2349 (2 + 3) × 4 + 9 = . 17658 1×7×6×5+8
•
2835 2 + 8 + 35 = . 69174 6 × (9 + 174)
•
1458 1 + 45 + 8 = . 32967 3 + 29 × 6 × 7
•
1836 18 + 3 × 6 = . 75429 7 × 5 × 42 + 9
•
2349 2+3+4+9 = . 78561 7 + 85 × (6 + 1)
•
2916 29 + 1 + 6 = . 34587 (3 + 4) × (5 + 8 × 7)
•
2916 2 × (9 + 1 + 6) = . 47385 4 × (7 + 3) × (8 + 5)
•
2916 29 + 1 + 6 = . 54837 5+4×8×3×7
•
2916 2 + 9 × 16 = . 74358 743 × 5 + 8
•
2 × (9 + 6 + 4) 2964 = . 37518 37 × (5 + 1 × 8)
•
2967 2 + (9 + 6 + 7) = . 41538 41 × (5 + 3) + 8
(1 + 4) × 7 + 6 1476 = . • 38952 3×8×9×5+2
1854 1 × 8 × (5 + 4) • = . 93627 9 + 3627
2394 2×3+9+4 • = . 58716 5 × 8 + 71 × 6
1476 1 + 47 + 6 • = . 52398 5 + 239 × 8
1864 1×8+6×4 • = . 25397 2 + (53 + 9) × 7
2394 2×3+9+4 • = . 67158 (6 + 7) × (1 + 5 × 8)
1485 1+4+8×5 = . 32967 32 + 967
•
1536 1+5+3×6 = . 49728 49 + 728
•
1×9+3+2 1932 = . 54786 5 + 4 × 7 × (8 + 6)
•
•
1935 (1 + 9) × 3 + 5 = . 48762 4 + 876 + 2
•
2496 2 × (4 + 9) + 6 = . 37518 37 × (5 + 1 × 8)
1 + 5 × (4 + 3) 1543 = . 67892 6 + 789 × 2
•
1938 1+9+3×8 = . 27645 2 + 7 × (64 + 5)
•
2538 2+5+3+8 = . 64719 6 × (4 + 71) + 9
•
3159 3 + 15 + 9 = . 27846 (2 + 7 × 8) × 4 + 6
•
1548 1+5+4+8 = . 37926 3 × 7 × (9 + 2 × 6)
•
(1 + 9) × 7 + 5 1975 = . 34286 3 × (428 + 6)
•
2574 2 × (5 + 7 × 4) = . 38961 38 + 961
•
3162 (3 + 1) × (6 + 2) = . 58497 (5 + 8 × 4) × (9 + 7)
•
1578 1×5×7+8 = . 69432 6 + 943 × 2
2136 2+1+3+6 = . 85974 8 × 59 + 7 + 4
•
2 + (5 + 8) × 3 2583 = . 69174 6 × (9 + 174)
•
•
31 + 68 3168 = . 94752 9 × 47 × (5 + 2)
•
3179 3 + 1 × 79 = . 82654 82 × (6 + 5 × 4)
•
3186 3 × (1 × 8) + 6 = . 47259 4 + 7 × (2 + 5) × 9
•
3249 3×2+4+9 = . 16587 1 × 6 + (5 + 8) × 7
26 + 73 2673 = . 15984 (1 + 5) × 98 + 4
•
3249 3×2+4+9 = . 16758 1 + 6 + 7 × (5 + 8)
•
•
1584 15 + 84 • = . 92736 92 × 7 × (3 + 6) 1593 1 × 5 × (9 + 3) • = . 28674 2 × (8 × 67 + 4)
2138 2+1+3+8 • = . 45967 (4 + 5 × 9) × 6 + 7
•
1638 1×6×3+8 = . 45927 (4 + 5) × 9 × (2 + 7)
•
•
1638 1×6×3+8 = . 74592 74 × (5 + 9 + 2)
•
1679 1+6+7+9 = . 32485 (3 + 2) × (4 + 85)
•
1692 1+6+9+2 • = . 58374 5 + 8 × (3 + 74)
•
1+7+3+4 1734 = . 65892 6 × 5 × (8 + 9 + 2)
•
2 × (5 + 8 × 3) 2583 = . 69741 6 × 9 × (7 × 4 + 1) 2589 2+5×8+9 = . 67314 6 × (7 × 31 + 4)
2+1+6×7 2167 = . 95348 9 × (53 × 4 + 8)
•
•
2178 21 + 7 × 8 = . 45936 4 + 5 × 9 × 36
•
2673 2×6+7+3 = . 54189 5 + (41 + 8) × 9
•
3276 3 × 2 × (7 + 6) = . 41958 41 + 958
•
2178 2×1×7+8 = . 65934 659 + 3 + 4
•
2673 2×6+7+3 = . 98415 9 × (84 + 1 + 5)
•
3 + 2 × (7 + 8) 3278 = . 15496 15 × 4 + 96
•
3287 3 + 28 + 7 = . 16954 16 + 9 × 5 × 4
•
3289 3×2+8+9 = . 41756 4 × (1 + (7 + 5) × 6)
•
3456 34 + 5 × 6 = . 12798 (1 + 2) × (7 + 9 × 8)
•
3465 3 × (4 + 6) + 5 = . 81972 819 + 7 + 2
•
3465 3 × (4 + 6) + 5 = . 87219 872 + 1 × 9
•
3492 3+4+9+2 = . 71586 71 × 5 + 8 + 6
2184 (2 + 1) × 8 + 4 = . • 53976 5 + 3 + 9 × 76
•
2187 2 + 18 + 7 = . 65934 6 × (5 × 9 × 3) + 4
2689 2 × 6 + 89 • = . 45713 4 + 571 × 3
•
2693 2 × (6 + 9 + 3) = . 45781 45 + 7 × 81
2197 2+1×9×7 = . 58643 5 × (86 × 4 + 3)
•
•
2316 2+3+1+6 = . 75849 (7 + 5) × 8 × 4 + 9
•
2781 2+7+8+1 = . 43569 4×3+5×6×9
•
2349 (2 + 3) × 4 + 9 = . 15876 15 × 8 + 76
•
2835 (2 + 8) × 3 + 5 = . 49167 4 + 9 × 1 × 67
1743 1 × 7 × (4 + 3) = . 25896 (2 + 5) × (8 + 96)
•
•
1748 1×7+4+8 = . 39652 39 × (6 + 5) + 2
•
1782 17 + 8 × 2 = . 53946 53 + 946
•
2158 2 × 1 × (5 + 8) = . 46397 46 × (3 + 9) + 7
•
2457 2 × (45 + 7) = . 13986 1 + 3 + 98 × 6
2716 2 × (7 + 1 × 6) = . 58394 (5 + 8) × (39 + 4)
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
3519 3+5+1×9 = . 78246 7 × (8 × (2 + 4) + 6)
•
4158 4+1+5+8 = . 36729 3 × (6 × 7 + 2 + 9)
•
4653 4 × (6 + 5) + 3 = . 81972 819 + 7 + 2
•
5832 5 × 8 + 32 = . 49167 4 + 9 × 1 × 67
•
3542 35 + 42 = . 17986 17 × (9 + 8 + 6)
•
4158 4+1×5×8 = . 62937 629 + 37
•
4653 4 × (6 + 5) + 3 = . 87219 872 + 1 × 9
•
5837 (5 + 8) × (3 + 7) = . 26491 2 × (6 × 49 + 1)
3564 3 + 5 + 64 = . 27819 2 + 7 × 8 × (1 + 9)
•
4218 4 + 218 = . 35967 3+5×9×6×7
•
4938 4+9+3+8 = . 61725 6 × (1 + 7 × (2 + 5))
•
•
5872 5 × (8 + 72) = . 13946 1 + 3 + 946
3571 3 × (5 + 7 + 1) = . 92846 9 × 28 × 4 + 6
•
4218 4×2×1+8 = . 65379 6 + 5 + 3 × 79
•
5913 5×9×1+3 = . 86724 8 × (6 × 7 × 2 + 4)
4257 42 + 57 = . 18963 (1 + 8 × 9) × 6 + 3
•
3612 3+6+1+2 = . 47859 47 + 8 × (5 + 9)
•
(5 + 9) × 3 + 4 5934 = . 17286 (1 + 7) × 2 × 8 + 6
•
6137 6+1+3+7 = . 42598 4 + 2 + (5 + 9) × 8
•
6137 6+1+3+7 = . 94582 9 × 4 × 5 + 82
5346 5×3×4+6 = . 12879 12 × 8 + 7 × 9
•
6138 6 + 1 × 38 = . 29574 2 × (95 + 7 + 4)
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5346 5 × 34 + 6 = . 17982 (1 + 7) × (9 × 8 + 2)
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6 + (1 + 8) × 7 6187 = . 32549 3+2×5×4×9
•
5346 5+3+4+6 = . 27918 2 × 7 + (9 + 1) × 8
•
6237 6×2+3×7 = . 15498 1+5+4+9×8
•
5346 5 + 3 + 46 = . 87219 872 + 1 × 9
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6237 62 + 37 = . 91854 9 × 18 × (5 + 4)
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5368 5+3+6×8 = . 12749 12 × 7 + 49
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6273 6 × (2 × 7 + 3) = . 15498 154 + 98
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6279 6 × 2 + 79 = . 14835 (1 + 4) × (8 + 35)
•
6×3×1+8 6318 = . 95742 (9 + 5) × 7 × 4 + 2
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6321 6×3+2+1 = . 47859 47 + 8 × (5 + 9)
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6 × (3 + 2) + 7 6327 = . 95418 9 × (54 + 1 × 8)
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6417 6×4×1+7 = . 23598 2 × (3 + 5) + 98
5712 5+7+1×2 = . 49368 49 + (3 + 6) × 8
•
6417 6×4×1+7 = . 28359 2 × 8 × (3 + 5) + 9
•
•
•
•
•
3 × (6 + 2) + 7 3627 = . 18954 (1 + 8 + 9) × (5 + 4) 3627 3 × (6 + 2) + 7 = . 45981 (4 + 5 × 9) × 8 + 1 3645 3 × (6 + 4 + 5) = . 12879 12 × 8 + 7 × 9
4329 4 + 3 × (2 + 9) • = . 15678 1 × 56 + 78 4 + 3 × (2 + 9) 4329 = . • 85176 8 × (5 × 17 + 6)
3654 3+6+5×4 = . 19278 1 × 9 × (2 + 7 + 8) •
•
3726 3+7×2+6 = . 18954 1 + 8 × (9 + 5) + 4 •
•
3+7×2+6 3726 = . 91854 9 × (1 + 8 + 54) •
3+7+4+9 3749 = . • 18256 1 × 82 + 5 × 6 3759 3 × (7 + 5 + 9) • = . 28461 2 × 8 + 461
•
•
3762 3×7+6×2 = . 51984 (5 + 1) × (9 × 8 + 4) 3+7+6×2 3762 = . 95418 9 × (54 + 1 × 8)
•
•
•
4356 43 + 56 • = . 81972 81 × (9 + 7 × 2)
• •
513 5 + 13 • = . 894672 8 × 9 × (4 + 6 × 72)
•
•
•
•
4368 4 + 3 × 68 = . 27951 2 × 7 × 95 + 1 4378 4+3+7+8 = . 29651 2 × (9 + 65) + 1 4392 43 + 9 × 2 = . 15768 1+5×7×6+8 4392 43 + 9 × 2 = . 17856 1+7+8×5×6 4 × (5 + 3 + 6) 4536 = . 12879 12 × 8 + 7 × 9 4536 4 + 53 + 6 = . 19728 19 × 7 × 2 + 8
5271 5 + 2 × 71 = . 96384 96 × (3 × 8 + 4) 5274 5 + 2 × (7 + 4) = . 36918 3 × (6 × 9 + 1 + 8)
5 × (3 × 8 + 2) 5382 = . • 61479 6 + 1479 5427 5 + 42 + 7 • = . 68139 6 × (8 × 13 + 9)
4563 45 + 63 = . 92781 9 + 27 × 81
•
4598 4+5×9+8 = . 23716 2 × 3 × 7 × (1 + 6)
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5472 5 × (4 + 7) + 2 = . 31968 319 + 6 + 8 5643 5 × 64 + 3 = . 27918 2 × (791 + 8)
3816 3 × (8 + 16) = . 52947 52 + 947
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4617 4 + 6 + 17 = . 29583 2 × 9 × 5 + 83
•
3876 3+8+7+6 = . 25194 2 × (5 + 1) × (9 + 4)
•
4623 (4 + 6) × 2 + 3 = . 71958 7 × (1 + 9) × 5 + 8
•
5728 (5 + 7) × 2 + 8 = . 61934 6 × 19 × 3 + 4
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6432 (6 + 4) × 3 + 2 = . 71958 7 × (1 + 9) × 5 + 8
•
3924 (3 + 9) × 2 + 4 = . 76518 7 × 6 × (5 + 1 × 8)
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4632 (4 + 6) × 3 + 2 = . 97851 9 × (7 + 8) × 5 + 1
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5814 5+8+1×4 = . 37962 3 × (7 + (9 + 6) × 2)
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6497 6+4+9×7 = . 21538 2 × (1 + 5 × 3 × 8)
•
3+9+4+2 3942 = . 61758 6 × 1 × (7 + 5 × 8)
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4+6×5×3 4653 = . 27819 2 + 7 × 8 × (1 + 9)
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5829 5×8+2×9 = . 43617 4 + 3 + 61 × 7
•
6534 6+5×3×4 = . 81972 819 + 7 + 2
•
•
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
6534 6+5×3×4 = . 87219 872 + 1 × 9
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8345 8+3+4+5 = . 21697 (2 + 1) × (6 + 9) + 7
•
9261 92 + 6 × 1 = . 45738 4 + (57 + 3) × 8
•
6578 65 + 78 = . 13294 1 + 3 × (2 + 94)
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8416 8 × 4 + 16 = . 95732 (9 + 5) × (7 + 32)
•
9261 92 + 6 × 1 = . 57834 578 + 34
•
6732 67 + 32 = . 19584 (19 + 5) × (8 + 4)
•
8421 84 × (2 + 1) = . 63759 6 × (3 + 7 × 5 × 9)
•
9367 9+3×6+7 = . 15428 1 + 5 + 42 + 8
6×8+7+3 6873 = . 51429 5 + 1 × 429
•
8514 85 + 14 = . 37926 3 × 7 × (9 + 2 × 6)
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8541 (8 + 5) × (4 + 1) = . 26937 2 × (6 + 93) + 7
•
•
•
•
6972 6 × (9 + 7) + 2 = . 15438 1 + (5 + 4) × 3 × 8 7128 7+1+2+8 = . 43956 4 × 3 + 9 × (5 + 6)
7128 7+1+2+8 • = . 93456 9 + 3 + 4 × 56
•
•
8576 8 × (5 + 76) = . 49312 4 × 931 + 2
8×6+9×4 8694 = . • 15732 1 + 5 + 73 × 2
•
•
261 2+6×1 = . 859473 8 × (5 × 94 × 7 + 3)
•
2+7+3 273 = . 645918 6 × (4 + 591 × 8)
•
274 2+7×4 = . 695138 6 + 9513 × 8
•
321 3 × (2 + 1) = . 697854 6 + 978 × 5 × 4
•
3×2+4 324 = . 179658 1 + 7 × 9 × (6 + 5) × 8
•
324 32 + 4 = . 196875 (19 + 6) × 875
9+3×6+7 9367 = . 45182 4 × (5 + 18 × 2) 9387 93 + 8 × 7 = . 65142 6 + 514 × 2 9+4+2+7 9427 = . 36851 (3 + 6 + 8) × 5 + 1
9472 9 + 472 • = . 13568 13 × (5 + 6 × 8)
•
7+1+8+2 7182 = . 94563 9 × (4 × 5 + 6) + 3
•
8712 8+7×1×2 = . 43956 4 × 3 + 9 × (5 + 6)
•
9×5+1+6 9516 = . 38247 (3 + 8 × 2) × (4 + 7)
•
3+5+1 351 = . 986427 9 + 86 × 42 × 7
•
7284 (7 + 2) × (8 + 4) = . 96513 9 × (6 + 51 × 3)
•
8712 8+7×1×2 = . 93456 9 + 3 + 4 × 56
•
9548 9 + 5 + 48 = . 23716 2 × (3 + 71) + 6
•
413 4×1×3 = . 285796 2 × 8 × (57 × 9 + 6)
•
7612 7×6×1+2 = . 35984 3 + 5 × (9 + 8 × 4)
•
8721 8+7+2×1 = . 56943 (5 + 6) × 9 + 4 × 3
•
9 × (5 + 7) + 6 9576 = . 12348 1 + 2 + 3 × 48
•
4+3+2 432 = . 158976 (1 + 5) × 8 × (9 × 7 + 6)
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7×6+1+4 7614 = . 39852 3 × (9 × 8 + 5 × 2)
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8+7×3+6 8736 = . 54912 5 × 4 × (9 + 1 × 2)
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9576 9+5+7×6 = . 12483 1 + 2 × (4 + 8) × 3
•
618 6+1×8 = . 495327 49 × (5 + 32 × 7)
7695 76 + 9 + 5 = . 12483 1 × 2 + 48 × 3
•
8932 8 × (9 + 3 + 2) = . 45617 4 + 561 + 7
•
9576 9 × (5 + 7) + 6 = . 23184 23 × 1 × (8 + 4)
•
684 6+8+4 = . 159372 (1 + 5 × 93) × (7 + 2)
•
693 6 + 93 = . 187425 (1 + 8) × 7 × 425
•
756 75 + 6 = . 123984 123 × 9 × (8 + 4)
•
•
•
7812 7 + 8 × (1 + 2) = . 39564 3 × (9 × 5 + 6) + 4 7832 78 + 32 = . 14596 1 + 4 × (5 × 9 + 6)
7 + (8 × 5 + 4) 7854 = . • 12936 1 + 2 + 9 × (3 + 6)
•
•
8957 8 + (9 + 5) × 7 = . 21463 2 + 1 × 4 × 63 9132 9+1+3×2 = . 86754 (8 + 6) × 7 + 54
•
•
9+6+1×4 9614 = . 72358 7 + (2 + 3 × 5) × 8 9675 9+6+7×5 = . 24381 2 + 43 + 81
9152 9 + 15 + 2 = . • 76384 7 + 6 × (3 + 8 × 4)
•
9734 9+7×3×4 = . 18526 1 + 85 × 2 + 6
•
7938 7+9×3+8 = . 16254 16 × 2 + 54
•
9153 9 × 1 × (5 + 3) = . 87462 8 × (74 + 6 × 2)
•
9752 9 + 75 × 2 = . 31648 3 + 1 + 64 × 8
•
948 9+4+8 = . 172536 1 × 72 × 53 + 6
•
7938 (7 + 9) × 3 + 8 = . 52164 52 × (1 + 6) + 4
•
9182 9+1+8+2 = . 73456 7 + 3 × (45 + 6)
•
216 2+1×6 = . 459378 4 + 5 × 9 × 378
•
972 9+7+2 = . 143586 1 + (435 + 8) × 6
•
8316 (8 + 3) × (1 + 6) = . 54972 5+4×9×7×2
•
9246 9+2×4+6 = . 31758 3 × 1 × 7 + 58
•
261 2+6+1 = . 539487 53 × 9 × (4 × 8 + 7)
•
972 9+7+2 = . 315846 3 + 1 × 5846
Inder J. Taneja
2.3
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
Fractions With Zero in Between
The above two subsections 2.1 and 2.2 brings results without digit 0. This subsection brings results where the digit 0 appears in between each fraction. In this case, the results are up to 10 digits.
•
102 10 + 2 = . 748 (7 + 4) × 8
105 1 × 05 • = . 378 3+7+8 105 1 × 05 • = . 924 (9 + 2) × 4
•
102 1 × 02 = . 6375 (6 × 3 + 7) × 5
1 × 02 102 = . • 7548 7×5×4+8
•
•
159 1×5+9 = . 2067 (20 + 6) × 7
168 1+6+8 • = . 9072 90 × (7 + 2)
102 1 × 02 = . 8976 8 × (9 + 7 + 6)
•
169 1 + 69 = . 3042 30 × 42
108 1 × 08 = . 243 (2 + 4) × 3
•
103 1+0×3 = . 6489 6 + 48 + 9
•
1+8+9 189 = . 2604 (2 + 60) × 4
•
108 1 + 08 = . 396 3×9+6
•
103 1 + 03 = . 8652 8 × 6 × (5 + 2)
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194 (1 + 9) × 4 = . 2037 20 × 3 × 7
•
127 1+2+7 = . 508 5 × 08
•
1+9+5 195 = . 208 2 × 08
•
•
1+0×4 104 = . • 9568 (9 + 5) × 6 + 8
•
209 2 + 09 = . 361 3×6+1
205 2 × 05 = . 1476 (1 + 4 + 7) × 6
•
105 10 + 5 = . 6384 6 × 38 × 4
•
207 2 + 07 = . 1495 (1 × 4 + 9) × 5
•
1+0×7 107 = . 2568 2 × 5 + (6 + 8)
207 2+0×7 • = . 3519 3 × 5 + 19
306 30 + 6 • = . 748 (7 + 4) × 8 306 3 + 06 = . 782 7+8×2
408 4 + 08 • = . 952 (9 + 5) × 2
1+0×7 107 • = . 5243 (5 + 2) × (4 + 3)
•
•
•
207 2 × 07 = . 4968 (4 × 9 + 6) × 8 207 2+0×7 = . 6831 6 × (8 + 3 × 1)
528 5+2×8 = . • 704 7 × 04
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109 1+0×9 = . 2834 2 × (8 + 3) + 4
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21 2+1 • = . 3906 (3 + 90) × 6
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127 1+2×7 = . 3048 30 × (4 + 8)
2+0×7 207 • = . 8694 8×6+9×4
42 4+2 = . 3906 (3 + 90) × 6
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84 8+4 = . 3906 (3 + 90) × 6
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142 (1 + 4) × 2 = . 3905 3 × 90 + 5
•
•
89 8+9 = . 3204 3 × 204
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14 + 7 147 = . 3906 (3 + 90) × 6
•
•
•
209 2 + 09 = . 3857 (3 × 8 + 5) × 7
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2+0×9 209 = . 5643 5 + 6 + 43
•
209 2 + 09 = . 6574 6 × 57 + 4
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209 2 + 09 = . 6745 (67 + 4) × 5
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215 (2 + 1) × 5 = . 6708 6 × (70 + 8)
•
(2 + 1) × 5 215 = . 9073 90 × 7 + 3
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216 2+1+6 = . 9408 (9 + 40) × 8
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234 2+3+4 = . 1508 1 × 50 + 8
•
234 2 × (3 + 4) = . 7605 7 × (60 + 5)
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239 2 × (3 + 9) = . 8604 860 + 4
•
243 (2 + 4) × 3 = . 6075 6 × 075
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2×5+8 258 = . 9073 90 × 7 + 3
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268 2 × (6 + 8) = . 1407 140 + 7
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268 2+6+8 = . 3015 30 × (1 + 5)
•
268 2+6×8 = . 7504 7 × 50 × 4
•
273 2+7+3 = . 4095 4 × 09 × 5
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284 2×8+4 = . 1065 10 + 65
•
284 2×8+4 = . 3905 3 × 90 + 5
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304 3 + 04 = . 7296 7 × (2 × 9 + 6)
•
305 3 × 05 = . 2196 2×1×9×6
207 2 + 07 = . 3864 3 × (8 + 6) × 4
108 1 + 08 = . 3456 (3 + 45) × 6
135 1×3×5 • = . 4608 (4 + 60) × 8
209 2 + 09 = . 1843 1+8×4×3
2 × 04 204 = . 7956 (7 + 9 × 5) × 6
•
2 × (8 + 9) 289 = . • 306 30 + 6
306 3 + 06 • = . 952 (9 + 5) × 2
•
105 1 × 05 = . 4368 4 + 3 × 68
•
•
104 1+0×4 = . 9672 9+6×7×2
204 2 × 04 • = . 1683 1 × 6 × (8 + 3)
•
207 2 + 07 • = . 9315 9 × 3 × 15 2 × 08 208 = . 1495 (14 + 9) × 5 2 × 08 208 = . 7956 (7 + 95) × 6
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
305 3 × 05 = . 4697 (4 × 6 + 9) × 7
•
405 40 + 5 = . 1368 (1 + 3 × 6) × 8
•
536 (5 + 3) × 6 = . 1809 18 × 09
•
702 7 × 02 = . 3861 (3 + 8) × (6 + 1)
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306 3 × 06 = . 7259 7 × (2 + 59)
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405 4 × 05 = . 2187 21 + 87
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539 5 + 39 = . 4018 (40 + 1) × 8
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702 7 + 02 = . 8346 83 + 4 × 6
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307 3+0×7 = . 4912 4 × (9 + 1 + 2)
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405 4 × 05 = . 2673 (2 + 6 × 7) × 3
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546 5×4+6 = . 1302 (1 + 30) × 2
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703 7 + 03 = . 5624 5 × (6 × 2 + 4)
•
307 3+0×7 = . 9824 9 × 8 + 24
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40 + 8 408 = . 1275 1 × 2 × 75
•
5×4+6 546 = . 2079 20 + 79
•
7 + 04 704 = . 1536 1+5+3×6
•
3 × 09 309 = . 5768 (57 + 6) × 8
•
409 4+0×9 = . 7362 7 + 3 + 62
•
562 (5 + 6) × 2 = . 3091 30 + 91
•
704 7 + 04 = . 6912 6×9×1×2
•
309 3 + 09 = . 8652 8 × 6 × (5 + 2)
•
423 4×2×3 = . 8601 8 × (60 + 1)
•
5×6×4 564 = . 3807 3 + 807
•
714 7 + 14 = . 3502 3 + 50 × 2
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315 3×1×5 = . 2079 20 + 79
•
426 4 + 26 = . 3905 3 × 90 + 5
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567 5+6+7 = . 4032 4 × 032
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726 (7 + 2) × 6 = . 9801 9 × (80 + 1)
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315 3×1×5 = . 2604 2 × 60 + 4
•
427 (4 + 2) × 7 = . 1098 10 + 98
•
576 5+7+6 = . 2048 2 × 04 × 8
•
729 72 + 9 = . 4608 (4 + 60) × 8
•
3×1×5 315 = . 6048 6 × 048
•
432 43 + 2 = . 1056 10 × (5 + 6)
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582 5×8+2 = . 1067 10 + 67
•
738 7+3+8 = . 1025 1 × (025)
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315 (3 + 1) × 5 = . 8064 8 × 064
•
4×3×2 432 = . 7056 7 × 056
•
5+9+2 592 = . 1036 10 + 3 × 6
•
7 × (4 + 2) 742 = . 6095 (60 + 9) × 5
•
321 3+2+1 = . 4708 (4 + 7 + 0) × 8
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501 5 + 01 = . 7682 76 + 8 × 2
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603 6 × 03 = . 2479 2 × (4 × 7 + 9)
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753 7 + 53 = . 6024 60 × 2 × 4
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324 3×2×4 = . 6075 6 × 075
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5 + 04 504 = . 1792 (1 × 7 + 9) × 2
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603 6 + 03 = . 2814 2 + 8 × (1 + 4)
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756 7+5+6 = . 3024 3 × 024
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3 × (2 + 4) 324 = . 7056 7 × 056
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504 5 + 04 = . 3192 3 × (1 + 9 × 2)
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6 × 03 603 = . 2948 2 × (9 × 4 + 8)
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7+9+2 792 = . 1408 1 × 4 × 08
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326 3 × (2 + 6) = . 4075 4 × 075
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504 5 + 04 = . 7896 (7 + 8) × 9 + 6
•
604 60 + 4 = . 1359 (1 + 3 × 5) × 9
•
801 8 + 01 = . 6497 6 + (4 + 9 × 7)
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3+2+7 327 = . 4905 4 × 9 × 05
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508 5 × 08 = . 1397 13 + 97
•
60 + 5 605 = . 2178 (2 + 1) × 78
•
801 8 + 01 = . 6942 6+9×4×2
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348 3 × (4 + 8) = . 7105 7 × 105
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516 (5 + 1) × 6 = . 9073 90 × 7 + 3
•
612 6×1×2 = . 5049 50 + 49
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8 + 01 801 = . 9345 (9 + 3 × 4) × 5
•
371 3×7×1 = . 6095 (60 + 9) × 5
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5+1+7 517 = . 9306 (9 + 30) × 6
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624 (6 + 2) × 4 = . 1053 1 + 053
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803 8 + 03 = . 1752 (1 × 7 + 5) × 2
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378 3+7+8 = . 2604 2 × 60 + 4
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519 51 + 9 = . 8304 8 × 30 × 4
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638 6 × (3 + 8) = . 7105 7 × 105
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803 8 + 03 = . 9271 9×2×7+1
•
387 3 + 87 = . 2064 20 × (6 × 4)
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527 5×2+7 = . 3069 30 + 69
•
651 6×5+1 = . 2079 20 + 79
•
804 8 + 04 = . 5293 52 + 9 × 3
•
392 (3 + 9) × 2 = . 1078 10 + 7 × 8
•
5+2×9 529 = . 7406 7 × (40 + 6)
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70 + 2 702 = . 1638 (1 + 6) × 3 × 8
•
805 8 × 05 = . 1932 1 + 93 + 2
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
806 8 + 06 = . 7254 7 × 2 × (5 + 4)
•
31 3×1 = . 94705 (9 + 4) × 705
•
1036 10 + 3 × 6 = . 2849 28 + 49
•
1089 1×0×8+9 = . 5324 (5 + 3 × 2) × 4
•
816 8 + 16 = . 3502 3 + 50 × 2
•
32 3×2 = . 17408 (1 + 7) × 408
•
1036 1 + 036 = . 4928 (4 + 9 × 2) × 8
•
1098 1+0×9+8 = . 5734 5×7+3×4
•
837 8+3+7 = . 2046 20 + 4 × 6
34 3+4 = . 15062 1 + 50 × 62
•
1038 1 + 03 + 8 = . 6574 6 × (5 + 7) + 4
•
•
(1 + 2 + 0) × 6 1206 = . 7839 78 + 39
•
837 83 + 7 = . 4092 40 × (9 + 2)
•
1208 1 × 2 × 08 = . 4379 (4 + 3) × 7 + 9
•
1208 1 × 2 × 08 = . 6795 6 + 79 + 5
•
1235 1×2+3+5 = . 7904 (7 + 9 + 0) × 4
•
(1 + 2 × 4) × 5 1245 = . 6308 6 × (30 + 8)
•
1246 (1 + 2 + 4) × 6 = . 5073 (50 + 7) × 3
1062 106 + 2 = . 3894 (3 + 8) × 9 × 4
•
(1 + 2 × 5) × 4 1254 = . 3078 30 + 78
847 (8 + 4) × 7 • = . 3025 30 × 2 × 5 8+6×1 861 = . • 3075 (3 + 07) × 5 902 9 + 02 • = . 3854 3+8×5+4 902 9 × 02 • = . 5863 (5 + 8) × (6 + 3)
•
•
•
•
•
•
9 + 06 906 = . 5134 5 × (13 + 4) 9×1×8 918 = . 2703 2 + 70 × 3
•
•
924 (9 + 2) × 4 = . 1806 1 × 80 + 6 926 9+2×6 = . 3704 3 × 7 × 04 927 9 × (2 + 7) = . 3605 (3 + 60) × 5
963 9+6+3 = . 4708 (4 + 7) × 08
54 5+4 = . 12096 (1 + 20) × 96 54 5+4 = . 37902 3 + 7 × 902
(1 + 04) × 5 1045 = . • 6897 68 + 97
•
•
1052 10 + 5 × 2 = . 8679 86 + 79 1056 1 × 05 × 6 = . 3872 38 + 72
7+1 71 = . 40896 (40 + 8) × 96
•
1023 10 + 23 = . 6758 6×7×5+8
•
•
1024 (10 + 2) × 4 = . 3968 3 × (9 × 6 + 8)
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1062 10 + 6 + 2 = . 4897 4+8×9+7
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1265 1 × (2 + 6) × 5 = . 7084 7 × 08 × 4
•
1026 1 + (026) = . 3458 (3 + 4) × (5 + 8)
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1064 1 × 064 = . 2793 2 × 7 × (9 + 3)
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1273 (12 + 7) × 3 = . 9045 9 × 045
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1026 1 × 02 × 6 = . 8379 8 + (3 + 7) × 9
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(1 + 06) × 4 1064 = . 9728 (9 + 7) × 2 × 8
•
1278 12 + 78 = . 3905 3 × 90 + 5
•
(1 + 2 + 8) × 7 1287 = . 4095 (40 + 9) × 5
•
1305 1 × 3 × 05 = . 7482 7 × (4 + 8) + 2
•
1308 1 + 3 + 08 = . 9265 (9 + 2 + 6) × 5
•
1309 13 + 09 = . 2856 2 + (8 × 5 + 6)
•
1309 1 + 3 × 09 = . 4862 (4 + 8 × 6) × 2
•
1364 1 × (3 + 6) × 4 = . 9207 9 × (20 + 7)
1085 1×0×8+5 = . 3472 3+4+7+2
•
1372 1 + 3 × (7 + 2) = . 5096 50 + 9 × 6
•
1372 1×3×7×2 = . 9408 9 × 4 × 08
•
1376 1 × 37 + 6 = . 2048 2 × 04 × 8
•
•
9 + 18 918 = . 3502 3 + 50 × 2
946 9 + 46 • = . 1032 10 × 3 × 2 •
4×1 41 = . • 58097 5 + 809 × 7
1057 (1 + 05) × 7 = . 2869 2 × (8 × 6 + 9)
1028 10 + 2 + 8 • = . 4369 4 + (3 + 6) × 9
1072 (1 + 07) × 2 • = . 6834 6+8×3×4
1028 1×0×2+8 • = . 7453 (7 + 4) × 5 + 3
1072 (1 + 07) × 2 = . • 8643 86 + 43
•
•
1029 10 + 2 + 9 = . 3675 3 + 6 × (7 + 5) 1032 10 × 3 × 2 = . 9546 9 + 546
•
•
1075 1 × 075 = . 4386 (43 + 8) × 6 1078 1 + 07 + 8 = . 2695 2 × (6 + 9 + 5)
1034 10 + 3 × 4 = . 6298 62 + 9 × 8
•
1035 1 × 03 × 5 = . 4278 4 + (2 + 7 × 8)
•
•
1035 1 × 03 × 5 = . 4692 4 × (6 + 9 + 2)
•
1085 (10 + 8) × 5 = . 6293 6 × 29 × 3
•
1035 1 × 03 × 5 = . 6279 6 × 2 + 79
•
1086 (1 + 08) × 6 = . 5249 (5 + 24) × 9
986 (9 + 8) × 6 • = . 1305 130 + 5
•
•
986 (9 + 8) × 6 = . 7105 7 × 105
•
•
1+8 18 = . 63504 63 × 504
•
21 2+1 = . 63504 6 × 3 × 504
1083 1×0×8+3 = . 7942 7+9+4+2
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
1386 13 + 86 = . 9072 9 × 072
•
1608 1 × 6 × 08 = . 2479 2 × (4 × 7 + 9)
•
1826 1+8+2×6 = . 7304 7 × 3 × 04
•
1958 1×9+5+8 = . 3204 32 + 04
•
1386 1+3+8+6 = . 9702 9 × 7 × 02
•
1608 16 + 08 = . 5293 52 + 9 × 3
•
1834 1+8+3×4 = . 2096 2 × 09 + 6
•
1962 1×9×6×2 = . 7085 (70 + 8) × 5
•
1407 14 + 07 = . 5293 52 + 9 × 3
•
(1 + 6) × 2 + 4 1624 = . 7308 73 + 08
•
1836 18 + 3 × 6 = . 5049 50 + 49
•
1976 (1 + 9) × 7 + 6 = . 5408 5 × 40 + 8
•
1407 14 + 07 = . 6298 6 + (2 + 9) × 8
•
1632 (1 + 6) × 3 × 2 = . 8704 8 × 7 × 04
•
1859 18 × 5 + 9 = . 2704 2 × 70 + 4
•
2016 20 + 16 = . 3584 (3 + 5 + 8) × 4
•
1408 14 × 08 = . 3256 3 + 256
•
1634 (1 + 6 × 3) × 4 = . 2709 2 × 7 × 09
•
(1 × 8 + 6) × 2 1862 = . 7049 70 + 4 × 9
•
2016 2 × 016 = . 9387 93 + 8 × 7
•
1425 (1 + 4) × 2 × 5 = . 3078 30 + 78
•
1+6×4+9 1649 = . 2037 2 × 03 × 7
•
1869 1+8+6×9 = . 5073 (50 + 7) × 3
•
(2 + 04) × 3 2043 = . 9761 9 + 76 + 1
•
1432 (1 + 4 + 3) × 2 = . 5907 59 + 07
•
1674 1+6+7+4 = . 5208 (5 + 2) × 08
•
1872 1×8×7×2 = . 4095 (40 + 9) × 5
•
2075 2 × 075 = . 3486 (34 + 8) × 6
•
1435 1+4+3×5 = . 6027 6 × 02 × 7
•
1683 (1 + 6 + 8) × 3 = . 2057 20 + 5 × 7
•
1876 1 × 8 + 76 = . 9045 9 × 045
•
2 × 07 × 9 2079 = . 3168 (3 + 1) × (6 × 8)
•
1×4×5+6 1456 = . 9072 90 + 72
•
1694 1×6+9×4 = . 3025 3 × 025
•
1896 1 + 89 + 6 = . 2054 20 × 5 + 4
•
2079 20 + 79 = . 4368 4 + 3 × 68
•
1485 1 + 4 + 85 = . 2079 2 × 07 × 9
•
1704 1 + 7 + 04 = . 5396 5+3×9+6
•
1903 19 + 03 = . 6574 6 × (5 + 7) + 4
•
2+0×7+9 2079 = . 5481 5×4+8+1
•
1503 (1 + 5) × 03 = . 7682 76 + 8 × 2
•
1×7+0×8 1708 = . 4392 4+3+9+2
•
1904 1 + 9 + 04 = . 3672 3×6+7+2
•
2086 2 × (08 + 6) = . 4917 49 + 17
•
1 + 5 + 06 1506 = . 9287 9×2+8×7
•
1724 (1 + 7 × 2) × 4 = . 5603 (5 + 60) × 3
•
19 + 08 1908 = . 2756 2+7+5×6
•
2 + 09 + 3 2093 = . 5681 5×6+8×1
•
1567 1 × 56 + 7 = . 9402 9 × (40 + 2)
•
1736 1+7×3+6 = . 5084 50 + 8 × 4
•
1926 1 + 9 + 26 = . 4708 (4 + 7) × 08
•
2105 (2 + 1) × 05 = . 9683 (9 + 6 + 8) × 3
•
1573 (1 + 5 + 7) × 3 = . 2904 2 × 9 × 04
•
1738 (1 + 7 + 3) × 8 = . 2054 20 × 5 + 4
•
1928 (1 + 9 × 2) × 8 = . 6507 6 + 507
•
21 × 07 2107 = . 4386 (43 + 8) × 6
•
1589 1×5×8+9 = . 2043 20 + 43
•
1792 (1 × 7 + 9) × 2 = . 6048 60 + 48
•
1932 1 + 9 × (3 + 2) = . 8064 8 × 06 × 4
•
2109 2 + 109 = . 3857 (3 × 8 + 5) × 7
•
1593 1 × 59 × 3 = . 4608 (4 + 60) × 8
•
(1 × 7 + 9) × 2 1792 = . 8064 80 + 64
•
1 + 9 + 35 1935 = . 2064 2 × 06 × 4
•
2109 2 + 109 = . 6574 6 × 57 + 4
•
1602 1 × 60 × 2 = . 5874 5 × 8 × (7 + 4)
•
1804 18 + 04 = . 3526 3 + 5 × (2 + 6)
•
1935 1×9×3×5 = . 6708 6 × (70 + 8)
•
2109 2 + 109 = . 6745 (67 + 4) × 5
•
1602 16 + 02 = . 9345 (9 + 3 × 4) × 5
•
(1 + 8) × 05 1805 = . 4693 4 × 6 + 93
•
1946 1 + 9 + 46 = . 2085 20 + 8 × 5
•
2139 (2 + 1) × 3 + 9 = . 5704 (5 + 7) × 04
•
1603 1 × 60 + 3 = . 2748 (2 + 7) × (4 + 8)
•
1806 1 × 8 + 06 = . 5934 (5 + 9) × 3 + 4
•
1953 1 × 9 + 53 = . 6048 6 × 04 × 8
•
2148 2 × 1 × (4 + 8) = . 5907 59 + 07
•
1 × 6 × 08 1608 = . 2345 2 × (3 + 4) × 5
•
1809 (1 + 8) × 09 = . 2546 2 × 54 + 6
•
1958 1×9+5+8 = . 2403 24 + 03
•
2+1+5+9 2159 = . 3048 3 × (0 × 4 + 8)
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
2173 2 + 17 × 3 = . 4059 40 + 59
•
2706 27 + 06 = . 3854 3+8×5+4
•
3016 (3 + 01) × 6 = . 9425 (9 + 4 + 2) × 5
•
2184 (2 + 1) × (8 + 4) = . 5096 (5 + 09) × 6
•
2709 2 × 7 × 09 = . 4386 4 × (3 + 8 × 6)
•
3017 (3 + 01) × 7 = . 9482 (9 × 4 + 8) × 2
•
2 × (7 + 0 × 9) 2709 = . 8514 8×5×1+4
2197 2+1×9×7 • = . 5408 5 × 4 × 08
•
•
2197 2 × (19 + 7) = . 6084 60 + 84 2 + 30 + 4 2304 = . 7168 (7 + 1 + 6) × 8
2714 2 + 7 + 14 • = . 3068 3 × 06 + 8 •
2315 2 × 3 × 15 • = . 7408 7 × 40 + 8
•
2349 234 + 9 • = . 7105 7 × 105
•
2365 2+3×6+5 • = . 8041 80 + 4 + 1 23 + 76 2376 = . • 9408 (9 + 40) × 8 2 + 37 + 6 2376 = . • 9504 9 × 5 × 04
•
2378 2 × (3 × 7 + 8) = . 4059 40 + 59
•
•
•
(2 + 38) × 4 2384 = . 6705 6 × (70 + 5) 2409 24 + 09 = . 8176 8 × (1 + 7 + 6)
•
•
•
2451 2 + 4 + 51 = . 3096 (3 + 09) × 6
•
3159 3 + 15 + 9 = . 6084 6 × 08 + 4
•
3186 (3 + 1 × 8) × 6 = . 9027 90 × 2 + 7
•
3192 3 × (1 + 9 × 2) = . 8064 80 + 64
•
3208 (3 + 2) × 08 = . 5614 5 + 61 + 4
•
3208 (3 + 2) × 08 = . 7619 76 + 19
•
3245 (3 + 2 + 4) × 5 = . 9086 9 × (08 + 6)
•
3 + 25 + 8 3258 = . 4706 4 × (7 + 06)
•
3405 3 × (40 + 5) = . 9761 9 × (7 × 6 + 1)
•
3 × (4 + 6) + 5 3465 = . 8019 (8 + 01) × 9
•
3465 3 × (4 + 6 + 5) = . 9702 9 × 7 × 02
3071 30 + 7 × 1 = . 9628 9×6×2+8
•
3465 3 × (4 + 6) + 5 = . 9801 98 + 01
•
(3 + 07) × 5 3075 = . 4182 4 × (1 + 8 × 2)
•
3 × (5 + 01) 3501 = . 4279 4+2+7+9
•
3078 3 + 078 = . 9614 9 + 61 × 4
•
3502 3 + 50 × 2 = . 4896 48 + 96
•
3504 3 × (50 + 4) = . 8176 (8 + 1) × 7 × 6
•
3516 (3 + 5 × 1) × 6 = . 8204 (8 + 20) × 4
•
3526 3 + 5 × (2 + 6) = . 4018 40 + 1 + 8
•
3567 (35 + 6) × 7 = . 4089 40 × 8 + 9
•
3645 3 × (6 + 4 + 5) = . 8019 80 + 19
•
3654 3 + 6 + 54 = . 9802 9 + 80 × 2
•
(3 × 6 + 5) × 7 3657 = . 8904 (8 + 90) × 4
3024 30 + 24 • = . 5768 5 + 7 × (6 + 8)
•
30 + 24 3024 = . 7896 (7 + 8) × 9 + 6
•
3042 3 × 04 × 2 = . 8619 8 + 6 × (1 + 9)
•
(3 + 04) × 5 3045 = . 8961 (8 + 9) × 6 + 1
2769 2 + 7 + 69 = . 3408 3 × 4 × 08 2781 27 × (8 + 1) = . 3605 (3 + 60) × 5 2816 2 × (8 + 1) + 6 = . 9504 9 × (5 + 04)
3048 3 × (0 × 4 + 8) • = . 9652 9 + 65 + 2
2849 2 × (8 + 4 + 9) = . 6105 6 × (10 + 5) 2864 2 × (8 + 6) + 4 = . 5907 59 + 07 (2 + 9 × 1) × 5 2915 = . 3074 30 + 7 × 4
•
•
2948 (2 × 9 + 4) × 8 = . 3015 30 × (1 + 5) 2951 2×9×5+1 = . 4086 40 + 86 2981 (2 + 9) × 8 × 1 = . 4065 4 × 06 × 5
3024 30 × (2 + 4) = . 5796 5 × (7 × 9 + 6)
•
2937 2 × 9 + 37 • = . 4806 4 + 80 + 6
2416 2 × (4 + 1) + 6 = . 8305 (8 + 3) × 05 2436 2 × (4 × 3 + 6) = . 7105 7 × (10 + 5)
3146 3+1×4+6 = . 5082 5 + 08 × 2
•
• •
•
•
• •
2754 2×7+5×4 = . 8019 80 + 19
3108 3 + (10 + 8) = . 5624 5×6+2×4
•
• •
2 + 7 + 43 2743 = . 5908 (5 + 9) × 08
•
3052 (30 + 5) × 2 = . 7194 71 + 94 30 + 7 × 1 3071 = . 8964 8 + 96 + 4
3 × (09 + 4) 3094 • = . 5712 (5 × 7 + 1) × 2 3096 (3 + 09) × 6 • = . 4257 42 + 57
2983 (2 + 9 + 8) × 3 = . 7065 70 + 65
•
3105 3 × (10 + 5) = . 4278 4+2+7×8
2508 2 × 50 + 8 = . 6479 (6 × 4 + 7) × 9
•
3015 3 × 015 = . 6298 6 + (2 + 9) × 8
•
•
2618 2 × (6 + 1 × 8) = . 5049 5 + 049
•
3015 3 × 015 = . 9246 92 + 46
•
3105 3 × (10 + 5) = . 6279 6 × 2 + 79
•
2×6+3×9 2639 = . 7105 7 × (10 + 5)
•
3016 30 + 1 × 6 = . 5278 5+2+7×8
•
(3 + 10) × 5 3105 = . 8694 (8 + 6) × (9 + 4)
•
3105 3 × (10 + 5) = . 4692 4 × (6 + 9 + 2)
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
3854 3+8×5+4 = . 9102 9 + 102
•
4098 4+0×9+8 = . 7513 7+5×1×3
•
4536 4 × (5 + 3 + 6) = . 8019 80 + 19
•
5298 5 + 2 × (9 + 8) = . 7064 (7 + 06) × 4
•
3872 3 + (8 + 7) × 2 = . 9504 9 × (5 + 04)
•
4102 4 × 10 + 2 = . 6739 6 × (7 + 3) + 9
•
4592 45 + 9 + 2 = . 8036 80 + 3 × 6
•
5302 53 + 02 = . 8194 (8 + 1) × 9 + 4
•
3892 3×8+9×2 = . 7506 75 + 06
•
4 × 10 + 2 4102 = . 9376 9 × (3 + 7) + 6
•
4653 4 × (6 + 5) + 3 = . 8019 (8 + 01) × 9
•
(5 + 3 × 2) × 4 5324 = . 9801 9 × (8 + 01)
•
3906 (3 + 90) × 6 = . 4872 4 × 87 × 2
•
4 × (1 + 2) × 8 4128 = . 7095 70 + 95
•
4 × (6 + 5) + 3 4653 = . 9801 98 + 01
•
5328 5+3+2×8 = . 7104 (7 + 1) × 04
•
3924 3×9×2×4 = . 7085 (70 + 8) × 5
•
4152 4 × (1 + 5) × 2 = . 7093 70 + 9 + 3
•
4692 4+6×9×2 = . 7038 7 × 03 × 8
•
5372 5+3×7+2 = . 9401 9 + 40 × 1
•
(3 + 9) × 4 + 1 3941 = . 5067 50 + 6 + 7
•
4163 4+1+6×3 = . 8507 8 × 5 + 07
•
4 + 8 + 07 4807 = . 6325 6×3+2+5
•
5402 5 + 402 = . 8176 8 × (1 + 76)
•
3951 3 × 9 × (5 + 1) = . 7024 (70 + 2) × 4
•
4+1+7×3 4173 = . 8025 (8 + 02) × 5
•
4809 4 + 8 + 09 = . 5267 5×2+6+7
•
5412 54 + 12 = . 8036 80 + 3 × 6
•
4016 4 × 016 = . 9287 92 + 8 × 7
•
4176 (4 + 1 + 7) × 6 = . 9802 9 + 80 × 2
•
4816 48 × (1 + 6) = . 9073 90 × 7 + 3
•
5724 5 + 7 + 24 = . 9063 9 × 06 + 3
•
4029 (40 + 2) × 9 = . 6715 6 × 7 × 15
•
4+1+8×5 4185 = . 9207 92 + 07
•
4 × (90 + 1) 4901 = . 5278 (5 + 2) × 7 × 8
•
5819 5 × 81 + 9 = . 6072 6 × 072
•
40 + 32 4032 = . 5768 5 + 7 × (6 + 8)
•
4208 4 × 2 + 08 = . 9731 9 + 7 × (3 + 1)
•
4908 4 × (9 + 0 × 8) = . 7362 (7 × 3 + 6) × 2
•
5819 5+8+1×9 = . 7406 7 × (4 + 0 × 6)
•
4032 40 × 3 × 2 = . 5796 5 × (7 × 9 + 6)
•
4×2+5×7 4257 = . 8019 (8 + 01) × 9
•
5049 5 + 049 = . 6732 6 × (7 + 3 + 2)
•
6293 6 + 29 × 3 = . 7105 7 × (10 + 5)
•
40 + 32 4032 = . 7896 (7 + 8) × 9 + 6
•
4257 4×2+5×7 = . 9801 98 + 01
•
5073 5 × 07 + 3 = . 9612 9 × (6 + 1 × 2)
•
6298 6 + (2 + 9) × 8 = . 7035 7 × 03 × 5
•
4035 40 + 35 = . 6187 6 × 18 + 7
•
4 + 3 × (2 + 9) 4329 = . 7605 (7 + 6) × 05
•
5082 50 + 8 × 2 = . 7469 7 + (4 + 6) × 9
•
6314 6 + 31 + 4 = . 9702 9 × (7 + 0 × 2)
•
4+0×6+9 4069 = . 5321 (5 + 3) × 2 + 1
•
4356 4×3×5×6 = . 9801 9 + 801
•
5083 (5 + 08) × 3 = . 7429 7 × 4 + 29
•
6325 6×3+2+5 = . 7084 7 × (0 × 8 + 4)
•
4069 4+0×6+9 = . 7512 7 + 5 + 12
•
4361 43 + 6 × 1 = . 7209 (7 + 2) × 09
•
51 + 03 5103 = . 8694 (8 + 6 + 9) × 4
•
6435 (6 + 4 + 3) × 5 = . 8019 (8 + 01) × 9
•
4069 4+0×6+9 = . 7825 7+8+2×5
•
4362 4 + 36 + 2 = . 5089 5 × 08 + 9
•
5124 (5 + 1 × 2) × 4 = . 6039 6 + 03 × 9
•
(6 + 4 + 3) × 5 6435 = . 9801 98 + 01
•
4075 40 + 7 × 5 = . 9128 (9 + 12) × 8
•
4376 4 × 3 + 76 = . 8205 8 × 20 + 5
•
5172 5 × 1 × 72 = . 6034 60 × (3 + 4)
•
6528 6×5×2+8 = . 7104 7 × 10 + 4
•
4081 4 × 08 + 1 = . 5936 (5 + 9) × 3 + 6
•
4386 4×3×8+6 = . 7095 70 + 95
•
5178 5 × 1 × (7 + 8) = . 6904 6 + 90 + 4
•
6532 65 + 3 × 2 = . 9108 91 + 08
•
4092 40 + 92 = . 6758 6×7×5+8
•
4503 4 + 50 + 3 = . 7268 7×2×6+8
•
(5 + 1) × (8 + 4) 5184 = . 9072 9 × 07 × 2
•
6534 6+5×3×4 = . 8019 (8 + 01) × 9
•
4095 (4 + 09) × 5 = . 6237 62 + 37
•
4509 45 + 09 = . 7682 76 + 8 × 2
•
5239 5 × 23 + 9 = . 6084 60 + 84
•
6579 6+5+7+9 = . 8041 8 × 04 + 1
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
6741 6×7×4×1 = . 8025 8 × 025
•
102 10 + 2 = . 47685 (4 + 7) × 6 × 85
•
106 1+0×6 = . 72398 (72 + 3) × 9 + 8
•
6924 6 × 9 × (2 + 4) = . 7501 7 × 50 + 1
•
102 1 × 02 = . 75684 7 × (5 + 6 × 8) × 4
•
•
7 × 03 × 5 7035 = . 9246 92 + 46
•
•
7035 7 × 03 × 5 = . 9648 96 + 48
•
•
7104 7 × 10 + 4 = . 8256 8×2×5+6
102 1+0×2 • = . 76398 7 × (6 + 3 + 98)
•
•
7128 7 + 128 = . 9504 9 × 5 × 04 •
(7 + 1) × 5 × 2 7152 = . • 8046 80 + 4 + 6
7612 76 + 12 • = . 8304 8 × 3 × 04 •
•
7 × (6 + 2 + 3) 7623 = . 8019 (8 + 01) × 9 7835 7+8+3×5 = . 9402 9 × (4 + 0 × 2)
•
18 + 9 189 = . 63504 6 × 3 × 504
•
196 1+9+6 = . 30478 (304 + 7) × 8
•
198 1 + 98 = . 63504 63 × 504
•
2 × 01 201 = . 76983 7 + 69 × (8 + 3)
•
204 2+0×4 = . 39678 3+9×6×7+8
•
204 2+0×4 = . 76398 7 × (6 + 3 + 98)
109 1+0×9 = . 26378 2 + 6 + 3 × 78
•
205 2 × 05 = . 63714 6 × 37 × 14
•
206 2 + 06 = . 51397 51 × 39 + 7
107 1+0×7 = . 29853 ((2 + 9) × 8 + 5) × 3 1 × 07 107 = . 45368 4 × 53 × (6 + 8) 108 10 + 8 = . 37296 3 × 7 × 296 108 1+0×8 = . 54972 5+4×9×7×2
•
104 1+0×4 = . 65728 (65 + 7 × 2) × 8
•
109 1+0×9 = . 27468 2 × 7 × (4 + 6 + 8)
•
105 1 × 05 = . 24738 (24 + 7) × 38
•
1 × 09 109 = . 62784 6 × 27 × 8 × 4
•
2 + 06 206 = . 84975 (8 + 4 × 9) × 75
•
105 1 × 05 = . 28476 (2 + 8 × 4 × 7) × 6
•
109 1+0×9 = . 64528 64 + 528
•
207 2 + 07 = . 13869 (1 + (3 + 8) × 6) × 9
1 × 05 105 = . • 63294 6 + 32 × 94
•
109 1+0×9 = . 73248 7 × 3 × (24 + 8)
•
207 2 + 07 = . 16583 1+6×5×8×3
105 1 × 05 • = . 63798 6 + 379 × 8
•
1×2×4 124 = . 90675 90 × (6 + 7) × 5
•
207 2+0×7 = . 19458 1×9×4×5+8
•
207 2 + 07 = . 39468 3 × (94 × 6 + 8)
•
208 2 × 08 = . 14365 (14 + 3) × 65
•
208 2 × 08 = . 17563 1 + 75 × 6 × 3
•
208 2+0×8 = . 53976 53 × 9 + 7 × 6
•
213 2+1+3 = . 67095 6 × 7 × 09 × 5
•
231 2+3+1 = . 46508 4 × 6 × 50 + 8
•
231 2×3+1 = . 74085 7 × 40 × 8 + 5
102 1 × 02 = . 34578 3 + 45 × (7 + 8)
•
102 10 + 2 = . 34986 (3 + 4) × 98 × 6
•
•
102 1+0×2 = . 35496 3 × (5 × 4 + 96)
•
•
102 1 × 02 = . 35649 3 × (56 × 4 + 9)
•
102 1+0×2 = . 39678 3+9×6×7+8
•
107 1+0×7 = . 25894 2 × ((5 + 8) × 9 + 4)
•
•
•
187 1×8+7 = . 35904 (3 + 5) × 90 × 4
104 1+0×4 = . 53976 53 × 9 + 7 × 6
8507 8 × 5 + 07 = . 9412 9 + 41 + 2
•
•
•
• 7406 7 × (40 + 6) = . 9315 9 × 3 × 15
7425 (7 + 4 × 2) × 5 • = . 9801 98 + 01
103 1 + 03 = . 84975 (8 + 4 × 9) × 75
•
107 1+0×7 = . 24396 2 × (4 × 3 × 9 + 6)
•
7302 7 × 3 × 02 • = . 8519 8×5×1+9
(7 + 4 × 2) × 5 7425 = . • 8019 (8 + 01) × 9
103 1+0×3 = . 29458 (2 × 9 + 4) × (5 + 8)
•
184 1×8+4 = . 20976 2 × 09 × 76
1 × 04 104 = . 39728 (3 × 9 × 7 + 2) × 8
•
•
10 + 3 103 = . 25647 2 + 5 × 647
•
105 10 + 5 = . 92736 9 × 2 × 736 1+0×6 106 = . 27984 2 × ((7 + 9) × 8 + 4)
132 (1 + 3) × 2 • = . 47058 4 × (705 + 8)
•
143 1+4×3 = . 70928 70 × 92 + 8 169 1×6+9 = . 30758 30 × 7 × (5 + 8)
106 1+0×6 = . 35298 35 + 298
•
106 1+0×6 = . 37842 3 × 7 + 8 × 42
•
•
106 1 × 06 = . 38425 (3 + 84) × 25
•
178 1+7+8 = . 36045 360 × (4 + 5)
•
106 1 × 06 = . 57293 5 × 72 × 9 + 3
•
1+7+8 178 = . 50463 504 × (6 + 3)
174 1+7+4 = . 30856 (30 + 8) × 56
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
237 2+3+7 = . 60514 60 × 51 + 4
•
304 3 × 04 = . 27968 2 × (7 × 9 + 6) × 8
•
308 3+0×8 = . 71456 (71 + 45) × 6
•
238 2×3+8 = . 10795 10 × 7 × 9 + 5
•
305 3 × 05 = . 19642 1 × 964 + 2
•
308 3+0×8 = . 91476 9 + 147 × 6
2×4+1 241 = . • 80976 8 × 09 × 7 × 6
•
241 2×4×1 = . 85073 8 × (50 × 7 + 3)
305 3 × 05 • = . 71248 (71 + 2) × 48
•
309 30 + 9 • = . 25647 2 + 5 × 647 3 × 09 309 = . 61285 (61 + 2) × 85
•
403 4+0×3 = . 51987 5 + (1 + 9 × 8) × 7
•
403 4 × 03 = . 82615 82 × 6 × 1 × 5
•
405 4 × 05 = . 19683 1 + 968 + 3
•
405 4 × 05 = . 73629 7 + 3629
•
405 4 × 05 = . 91368 (91 + 3) × 6 × 8
3 × 06 306 = . 12495 (1 + 2) × (49 × 5)
•
306 3+0×6 = . 12954 1 + 2 × (9 + 54)
•
3×1×5 315 = . 47082 4 × 70 × 8 + 2
•
405 40 + 5 = . 91728 91 × 7 × 2 × 8
241 2×4+1 = . 90375 9 × 0375
•
•
247 2+4+7 = . 10963 10 + 9 × 63
•
3 × 06 306 = . 17289 (1 + 7 × 2 × 8) × 9
•
315 3×1×5 = . 90867 90 × 8 × 6 + 7
•
406 4 + 06 = . 12789 (1 × 27 + 8) × 9
•
248 2×4+8 = . 90675 90 × (6 + 7) × 5
•
306 3 × 06 = . 19584 (1 + 95) × (8 + 4)
•
3 × (2 + 9) 329 = . 14805 1480 + 5
•
406 4+0×6 = . 19285 (1 + 9 + 28) × 5
•
(2 + 4) × 9 249 = . 10375 10 × 3 × 75
306 3+0×6 = . 29784 (2 + 9 × 7 + 8) × 4
•
361 3×6+1 = . 27094 2 × (709 + 4)
•
•
4+0×6 406 = . 23751 (2 + 37) × (5 + 1)
•
406 4+0×6 = . 35728 (35 + 7 + 2) × 8
•
407 4 + 07 = . 13986 1 × 3 × 9 × (8 + 6)
•
4 + 07 407 = . 25863 2 × 58 × 6 + 3
•
407 4+0×7 = . 95238 9 × (5 × 2 + 3) × 8
•
4+0×8 408 = . 29376 2 × 9 × (3 + 7 + 6)
•
249 2+4+9 • = . 30876 30 × (8 × 7 + 6)
•
(2 + 5) × 3 253 = . 67804 67 × (80 + 4)
306 3+0×6 • = . 49572 (49 + 5) × (7 + 2)
•
374 3 + 74 • = . 15062 1 + 50 × 62
3 × 06 306 = . 58429 5 + 8 × 429
•
376 (3 + 7) × 6 = . 10528 105 × 2 × 8 3+8×1 381 = . 20574 20 + 574
256 (2 + 5) × 6 = . 17408 1 × 7 × 408
•
306 3+0×6 = . 95472 9 × (5 + 47) × 2
•
•
2+6+1 261 = . 74095 7 × (40 × 9 + 5)
•
3 + 07 307 = . 12894 (12 × 8 + 9) × 4
•
384 3 × (8 + 4) = . 12096 (1 + 20) × (9 × 6)
•
284 2 × (8 + 4) = . 30175 30 × 17 × 5
•
307 30 + 7 = . 15964 (1 + 5 × 96) × 4
•
387 3 × (8 + 7) = . 40592 40 × 59 × 2
•
408 4+0×8 = . 67932 6 × (79 + 32)
•
285 2+8+5 = . 10963 10 + 9 × 63
307 3+0×7 • = . 19648 (1 + 9 + 6) × (4 + 8)
•
398 3+9+8 = . 70645 (706 + 4) × 5
•
409 4+0×9 = . 68712 6×8×7×1×2
402 4 × 02 = . • 31758 (3 + 1 + 75) × 8
•
4+1×3 413 = . 20768 (2 + 07 × 6) × 8
•
413 4+1×3 = . 26078 (2 + 60) × 7 + 8
•
415 (4 + 1) × 5 = . 30876 30 × (8 × 7 + 6)
•
418 4 + 18 = . 37905 3 × 7 × (90 + 5)
•
426 (4 + 2) × 6 = . 30175 30 × 17 × 5
•
4×3+7 437 = . 10925 (10 + 9) × 25
•
2+9+1 291 = . • 48306 4 × 83 × 06 297 2+9+7 • = . 35046 (350 + 4) × 6
•
•
3 + 08 308 = . 16492 1 + 6 × 49 × 2 308 3 + 08 = . 21756 21 × (7 + 5 × 6)
•
308 3 × 08 = . 21945 2 × 19 × 45
•
402 40 + 2 = . 37185 3 × 7 × 185
301 3 × 01 = . 97825 97 × (8 + 2) + 5
•
•
302 3 + 02 = . 57984 5 × ((7 + 9) × (8 + 4))
•
308 3 + 08 = . 59472 59 × 4 × (7 + 2)
•
402 4+0×2 = . 76983 7 + 69 × (8 + 3)
•
3+0×2 302 = . 74896 (74 + 8) × 9 + 6
•
308 3 × 08 = . 67914 6 × 7 × 9 × 14
•
4+0×3 403 = . 26598 2 × 6 × (5 + 9 + 8)
•
402 4+0×2 = . 68139 6 × (8 × 13 + 9)
Inder J. Taneja
•
452 4×5×2 = . 18306 (1 + 8) × 30 × 6
453 (4 + 5) × 3 • = . 10872 (1 + 08) × 72
•
461 4+6×1 = . 37802 (3 + 7) × (80 + 2)
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
506 5 + 06 • = . 13248 1 × 3 × 2 × 48
•
506 5 + 06 = . 13294 1 + 3 × (2 + 94)
•
594 5+9+4 = . 62073 (620 + 7) × 3
603 6 × 03 • = . 18425 (18 + 4) × 25
•
6 + 03 603 = . 21574 2 × (157 + 4)
•
627 6+2×7 = . 15048 15 × 04 × 8
•
639 6+3×9 = . 21087 2 + 1087
•
645 6+4+5 = . 23908 2 × (3 × 90 + 8)
•
645 6+4+5 = . 31089 3 + 10 × 8 × 9
•
(6 + 4) × 8 648 = . 10935 10 × 9 × 3 × 5
506 5 + 06 = . 13984 (1 + 3 + 9 × 8) × 4
•
•
506 5 + 06 = . 23184 2 × 3 × 1 × 84
•
603 6 × 03 = . 29547 2 × (9 + 54) × 7
•
649 6×4+9 = . 10325 (103 + 2) × 5
462 (4 + 6) × 2 = . 15708 (15 + 70) × 8
•
5+0×6 506 = . 29348 2 × 9 + 34 × 8
•
6 + 03 603 = . 29815 ((2 + 9) × 8 + 1) × 5
•
651 6×5+1 = . 47082 4 × 70 × 8 + 2
4 × (7 + 8) 478 = . 12906 (1 + 2) × 90 × 6
•
5+0×6 506 = . 81972 (81 + 9) × (7 + 2)
•
603 60 + 3 = . 41875 (4 + 1) × 875
•
6+8+1 681 = . 90573 (90 + 5) × 7 × 3
•
693 6+9×3 = . 10584 (1 + 05) × 84
•
6+9×3 693 = . 47082 4 × 70 × 8 + 2
•
693 6+9×3 = . 51408 51 × (40 + 8)
•
701 7 × 01 = . 86924 8×6×9×2+4
462 4×6+2 = . 10395 (10 + 3) × 9 × 5
•
•
462 (4 + 6) × 2 = . 10857 10 × (8 × 5 + 7)
•
•
•
504 5+0×4 = . 89712 89 × (7 + 1 + 2)
603 6+0×3 = . 29145 29 × (1 + 4 + 5)
481 4 × (8 + 1) • = . 36075 36 × 075
507 5 + 07 • = . 46982 (4 + 69 × 8) × 2
604 6 × 04 • = . 28539 (2 + 8 × 5) × 3 × 9
(4 + 8) × 2 482 = . • 90375 90 × (3 + 7) × 5
5+0×8 508 = . • 14732 (1 + 4 × 7) × (3 + 2)
604 6+0×4 • = . 32918 3 + 2 × 9 × 18
482 4×8+2 = . 97605 9 × (760 + 5)
•
•
4+8+6 486 = . 13905 (13 + 90) × 5
•
6+0×7 607 = . 81945 (8 + 1 + 9) × 45
509 5 × 09 = . 41738 (4 + 1) × 738
•
•
516 5+1+6 = . 23908 2 × (3 × 90 + 8)
•
609 6+0×9 = . 15428 (1 + (5 + 4) × 2) × 8
•
704 7 + 04 = . 13568 (1 + 3) × (5 + 6 × 8)
489 (4 + 8) × 9 = . 20375 20 × 3 × 75
•
517 5 × (1 + 7) = . 40326 40 × 3 × 26
•
609 6+0×9 = . 15834 1 × (5 + 8) × 3 × 4
•
704 7 + 04 = . 16832 1 + 6 + 8 × 32
•
502 5 × 02 = . 36897 (3 + 6 × (8 + 9)) × 7
•
521 5×2×1 = . 64083 6 + 408 × 3
•
609 6+0×9 = . 23548 (2 + 3 × (5 + 4)) × 8
•
709 7+0×9 = . 24815 ((2 + 4) × 8 + 1) × 5
•
502 5+0×2 = . 79316 79 × (3 + 1 + 6)
5 × 24 524 • = . 38907 3 + 8907
6+0×9 609 = . 23751 (2 + 37) × (5 + 1)
•
•
712 (7 + 1) × 2 = . 36045 3 × 6 × 045
•
7 + 14 714 = . 25908 2 + (5 + 90) × 8
•
721 7+2×1 = . 98056 9 × (80 + 56)
•
723 (7 + 2) × 3 = . 69408 6 × 9 × (40 + 8)
•
5+0×3 503 • = . 29174 2 + 9 × (1 + 7) × 4
524 5×2×4 • = . 90783 90 × 7 × (8 + 3)
504 50 + 4 = . 37296 37 × 2 × 9 × 6
•
504 5 + 04 = . 69832 (6 + 9) × 83 + 2
•
•
504 5 + 04 = . 79632 79 × (6 + 3) × 2
•
5 + 04 504 = . 86912 8 × (6 + 91) × 2
•
•
•
609 6+0×9 = . 35728 (35 + 7 + 2) × 8
528 5×2+8 = . 74096 7 × 40 × 9 + 6
•
531 5+3+1 = . 20768 (2 + 07 × 6) × 8
•
617 6 × (1 + 7) = . 32084 3 × (208 × 4)
•
729 7 + 29 = . 45603 4 × (560 + 3)
•
531 5+3+1 = . 26078 (2 + 60) × 7 + 8
•
621 6 + 21 = . 78039 (7 + 80) × 39
•
751 7×5+1 = . 63084 (6 + 30) × 84
•
562 (5 + 6) × 2 = . 10397 10 + 397
•
624 6×2+4 = . 73905 7 × 3 × 90 + 5
•
(7 + 5) × 2 752 = . 39104 (3 + 9) × 104
615 6×1×5 = . 29807 2 × (9 × 80 + 7)
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
754 (7 + 5) × 4 = . 39208 (3 + 9) × 208
•
837 8 + 37 = . 10695 (106 + 9) × 5
•
952 (9 + 5) × 2 = . 14076 1 + 407 + 6
•
1036 1 + 036 = . 59472 59 × 4 × (7 + 2)
•
782 (7 + 8) × 2 = . 50439 5 × 043 × 9
•
843 (8 + 4) × 3 = . 21075 (2 + 10) × 75
•
958 9+5×8 = . 21076 2 + 1076
•
1036 1+0×3+6 = . 94572 9 + 45 × 7 × 2
•
792 (7 + 9) × 2 = . 15048 150 × 4 + 8
•
8+4×6 846 = . 19035 1 × 90 × (3 + 5)
•
9+6+1 961 = . 80724 8 × 07 × 24
•
1042 10 + 4 + 2 = . 96385 (96 × 3 + 8) × 5
•
962 9+6×2 = . 30784 3 × 07 × 8 × 4
•
1+0×4+3 1043 = . 79268 ((7 + 9) × 2 + 6) × 8
•
972 9+7+2 = . 10368 (1 + 03) × 6 × 8
•
1045 1×0×4+5 = . 37829 37 + 8 × 2 × 9
•
31 3×1 = . 408952 408 × (95 + 2)
•
1×0×4+5 1045 = . 87362 8 × 7 + 362
4+2 42 = . • 169057 1 + 690 × 5 × 7
•
1047 1 + 04 + 7 = . 58632 5 × 8 + 632
•
1052 1+0×5+2 = . 96784 (9 × 6 + 7 + 8) × 4
•
1 + 05 + 3 1053 = . 27846 (2 + 7 × 8) × 4 + 6
•
1×0×5+4 1054 = . 67983 6×7+9×8×3
•
1056 1 + 05 + 6 = . 27984 2 + (7 + 9 × 8) × 4
•
1056 10 + 56 = . 37248 (3 + 72 × 4) × 8
•
1 + 05 + 8 1058 = . 49726 4 + 9 × 72 + 6
•
1059 1×0×5+9 = . 32476 3 × 2 × (4 + 7 × 6)
1034 10 + 3 × 4 = . 27589 2 + (7 + 58) × 9
•
1059 1×0×5+9 = . 62834 6 × (2 + 83 + 4)
1034 10 + 3 × 4 = . 59267 (5 + 92) × (6 + 7)
•
1064 (1 + 06) × 4 = . 23598 23 + 598
792 (7 + 9) × 2 • = . 53064 (530 + 6) × 4
•
•
•
801 8 + 01 = . 32574 3 × 2 × (57 + 4) 801 8 + 01 = . 36579 3 + 6 × (5 + 7 × 9) 803 8 + 03 = . 25769 2 + 57 × 6 + 9
863 8+6+3 • = . 50917 (50 + 9) × 17
•
•
•
864 8+6+4 = . 10752 (107 + 5) × 2 8+6×4 864 = . 13905 (13 + 90) × 5 864 8+6+4 = . 15072 (150 + 7) × 2
• 8 + 03 803 = . • 47596 4 + (7 + 5) × 9 × 6
•
803 8 + 03 = . 67452 6 × 7 × (4 × 5 + 2)
•
804 8 + 04 = . 27135 27 × 1 × 3 × 5
•
•
•
•
•
804 8 + 04 = . 72695 72 × (6 + 9) + 5 806 8 × 06 = . 42315 4 × 2 × 315
8+0×7 807 = . • 14526 14 + 5 × 26 8 + 12 812 = . • 59073 5 × (90 + 7) × 3
•
824 8+2×4 = . 30591 3 + 0591
•
867 8+6+7 = . 23409 (23 + 40) × 9 867 8+6+7 = . 34102 (3 + 410) × 2 8+7+3 873 = . 20564 (20 × 5 + 6) × 4 897 8+9+7 = . 10465 (10 + 46) × 5
9 + 02 902 = . • 37146 3 + (71 + 4) × 6
•
•
•
904 9 × 04 = . 87236 (8 × 72 + 3) × 6 905 9 × 05 = . 23168 (23 + 1) × 6 × 8
84 8+4 = . 169057 1 + 690 × 5 × 7
1024 1 × 02 × 4 • = . 35968 3+5×9×6+8 1026 102 + 6 = . • 35948 (3 + 5 × 94) × 8
•
•
•
•
•
1026 1 + 0 × 26 = . 54378 5 + 4 × (3 + 7) + 8 (1 + 02) × 8 1028 = . 63479 6 × (34 × 7 + 9) 1 + 03 + 2 1032 = . 59684 5 + 9 × (6 + 8 × 4)
9 × 05 905 = . 37648 3 × (7 + 6) × 48
•
1034 1 × 03 × 4 = . 69278 6×9×2×7+8
•
1065 1×0×6+5 = . 42387 4×2×3×8+7
826 8 × (2 + 6) = . 30975 30 × (9 + 7) × 5
•
906 9+0×6 = . 18724 18 + 7 × 24
•
1035 10 + 35 = . 67482 6 × (7 + 482)
•
1069 1×0×6+9 = . 24587 24 × 5 + 87
•
832 (8 + 3) × 2 = . 17056 1 + (70 + 5) × 6
•
908 9 + 08 = . 15436 1 + (5 + 43) × 6
•
1035 1×0×3+5 = . 78246 7 × (8 × (2 + 4) + 6)
•
1072 (1 + 07) × 2 = . 93465 93 × (4 + 6 + 5)
•
834 (8 + 3) × 4 = . 27105 2 × (710 + 5)
•
913 (9 + 1) × 3 = . 60258 60 × (25 + 8)
•
1036 10 + 3 × 6 = . 54279 54 × 27 + 9
•
1079 10 + 7 + 9 = . 28635 2 + 86 × (3 + 5)
•
8 + 36 836 = . 27094 2 × (709 + 4)
•
924 9×2+4 = . 15078 1 + 50 × 7 + 8
•
1036 1 + 03 + 6 = . 57498 5 × (7 + (4 + 9) × 8)
•
10 + 7 + 9 1079 = . 58432 (5 × 8 + 4) × 32
•
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
1089 (1 + 08) × 9 = . 24563 (24 + 5) × 63
•
1302 1 + 3 + 02 = . 69874 6 + (9 × 8 + 7) × 4
•
1432 (1 + 4 + 3) × 2 = . 59607 59 + 607
•
1598 (1 + 5 + 9) × 8 = . 60724 60 × (72 + 4)
•
1089 1 + 089 = . 53724 5 × 37 × 24
•
1302 1 + 3 × 02 = . 74586 7 × 45 + 86
•
1436 1×4+3×6 = . 50978 5 + 097 × 8
•
1605 (1 + 6) × 05 = . 82497 (8 + 249) × 7
1×4+3×8 1438 = . • 97065 9 × 7 × 06 × 5
•
1608 1 × 6 × 08 = . 29547 2 × (9 + 54) × 7
•
1608 1×6+0×8 = . 34572 3 + (4 + 5) × 7 × 2
•
1624 16 + 24 = . 59073 5 × (90 + 7) × 3
•
1629 1 + 62 + 9 = . 43078 (4 + 30) × 7 × 8
•
1638 1×6×3+8 = . 74025 (7 + 40) × 25
1092 10 + 9 × 2 • = . 43875 (4 + 3 + 8) × 75
•
1094 1×0×9+4 = . 62358 6 × (2 × 3 × 5 + 8)
1302 (1 + 30) × 2 • = . 75684 75 × 6 × 8 + 4
•
1×3+0×2 1302 = . 79856 (7 + 9) × 8 + 56
1456 1+4+5+6 • = . 20839 20 × (8 + 3) + 9 1467 1 × 4 × (6 + 7) = . 35208 3 × 52 × 08
1304 1 + 3 + 04 = . 72698 7×2+6×9×8
•
1 × 30 + 5 1305 = . 64728 6 × 4 × 72 + 8
•
•
1309 13 + 09 = . 27846 2 × (7 + 8 × 4) × 6
•
1 × 5 × 04 1504 = . 76328 7 + 63 × 2 × 8
•
1652 1+6+5+2 = . 30798 3 × (079 + 8)
1 + 2 + 04 1204 = . • 93568 (9 + 3 + 56) × 8
•
1 + 3 × 09 1309 = . 45628 (4 × 5 × 6 + 2) × 8
•
1508 1 × 50 + 8 = . 23764 2 + 3 × 76 × 4
•
(1 + 6) × 5 + 3 1653 = . 24708 2 × 4 × 70 + 8
1206 12 × 06 • = . 87435 87 × 4 × 3 × 5
•
1309 13 + 09 = . 62475 (6 + 24) × 7 × 5
•
1508 1 × 5 + 08 = . 93264 (9 + 32 × 6) × 4
•
1×6×5+7 1657 = . 43082 4 × 30 × 8 + 2
•
1672 1×6+7×2 = . 95304 95 × 3 × 04
•
1674 1+6+7×4 = . 80352 80 × 3 × (5 + 2)
•
(1 + 6) × 8 + 2 1682 = . 74095 7 × (40 × 9 + 5)
1547 1+5+4+7 = . 20839 20 × (8 + 3) + 9
•
1×6+8+3 1683 = . 57024 570 + 2 + 4
•
1683 1×6+8+3 = . 75042 750 + 4 × 2
•
1704 1 + 7 + 04 = . 29536 2 × (95 + 3 + 6)
•
1704 (1 + 70) × 4 = . 29568 (2 + 9) × 56 × 8
•
1704 1+7+0×4 = . 32589 3 × (2 + 5 × 8 + 9)
•
1708 1 + 7 + 08 = . 29463 (2 + 9 × (4 + 6)) × 3
•
1708 1+7×0×8 = . 49532 4 + 9 + (5 + 3) × 2
•
1708 1×7+0×8 = . 53924 5+3×9×2×4
•
1+7+2×8 1728 = . 34056 (3 + 40) × (5 + 6)
1094 1 × 09 × 4 = . 68375 (6 + 8 × 3) × 75
•
•
1+0×9+8 1098 = . 25376 2 × (5 + 3) × (7 + 6)
•
•
1203 1+2+0×3 = . 78596 7 × (8 + 5 + 9 + 6)
•
1206 12 × 06 = . 97485 97 × (4 + 8) × 5
1×3+5×2 1352 = . • 40976 4 × 097 + 6
1+2+3×7 1237 = . 89064 8 × 9 × 06 × 4
•
•
1243 1×2×4+3 = . 80795 (80 + 7 × 9) × 5
•
•
(12 + 4) × 5 1245 = . 38097 3 × (809 + 7)
1248 12 + 48 • = . 30576 (30 + 5) × 7 × 6
•
•
•
•
1248 1 × 24 + 8 = . 73905 7 × 3 × 90 + 5 1254 1×2+5+4 = . 76038 7 + 60 × (3 + 8)
1485 1+4+8×5 = . 62073 (620 + 7) × 3
1+5+3×2 1532 = . • 67408 6 × (7 + 4) × 08 1534 1 + 5 × (3 + 4) = . 82069 (8 + 206) × 9
1352 1×3+5×2 = . 80496 (80 + 49) × 6
•
1365 1+3+6+5 = . 20748 20 × (7 + 4) + 8
•
•
1368 1 + 3 + 68 = . 27094 2 × (709 + 4)
•
1548 (1 + 5 + 4) × 8 = . 23607 2 × (3 + 607)
•
1×3+7+6 1376 = . 40592 4 × 059 × 2
•
1 × 5 × (4 + 8) 1548 = . 60372 60 × (37 + 2)
1386 1 × 38 + 6 • = . 47502 (4 + 750) × 2 1386 13 + 8 + 6 • = . 57904 (5 + 7) × (90 + 4)
1586 1+5+8×6 • = . 30927 (30 + 9) × 27
•
1593 15 + 9 + 3 = . 20768 (2 + 07 × 6) × 8
1389 1×3×8+9 = . 24076 2 × (40 × 7 + 6)
•
•
1403 1 × 4 × 03 = . 72956 (7 + 2 + 95) × 6
•
1596 1+5+9×6 = . 20748 20 × (7 + 4 × 8)
•
14 + 0 × 7 1407 = . 38592 3 × 8 × (5 + 9 + 2)
•
1596 1 + 5 × (9 + 6) = . 47082 4 × 70 × 8 + 2
•
1257 1×2+5×7 = . 98046 9 × 80 × 4 + 6
•
•
1264 1×2×6×4 = . 57038 57 × 038
•
1298 1×2+9+8 = . 35046 3 + 504 + 6
1593 15 + 9 + 3 = . 26078 (2 + 60) × 7 + 8
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
1729 1+7+2+9 = . 58604 5 × 8 + 604
•
1853 1+8+5+3 = . 64092 6 × (40 + 9) × 2
•
1974 19 + 7 × 4 = . 32508 3 × (250 + 8)
•
1734 17 + 34 = . 25908 2 + (5 + 90) × 8
•
1859 1×8+5+9 = . 43602 43 × 6 × 02
•
2097 2+0×9+7 201 2 × 01 • = . = . 36814 3 × 6 × 8 + 14 475968 (4 + 7 × (5 + 9) × 6) × 8
1739 1+7+3+9 • = . 26085 260 + 8 × 5
•
•
(17 + 4) × 3 1743 = . 26809 2 × 6 × 80 + 9 1746 1+7+4+6 = . 83905 (83 + 90) × 5
1862 (1 + 8 × 6) × 2 • = . 37905 3 × 7 × (90 + 5)
•
•
1864 1 × (8 + 6) × 4 = . 92035 920 × 3 + 5
2016 2 × (01 + 6) • = . 37584 3 × (75 + 8 + 4)
•
1872 1 × 8 + 72 = . 65403 65 × (40 + 3)
•
2 × 03 + 6 2036 = . 75841 7 × 58 + 41
•
1+8+9+2 1892 = . 34056 3 × 4 × 05 × 6
•
2043 20 + 4 + 3 = . 16798 1 + (6 + 7) × (9 + 8)
•
1769 (1 + 7) × (6 + 9) = . 28304 2 × 8 × 30 × 4
•
1 × 9 × 02 1902 = . 38674 3 × (8 × 6 + 74)
•
20 + 4 + 9 2049 = . 75813 (7 × 58 + 1) × 3
•
1 × (7 + 8) × 2 1782 = . 43065 4 × 30 × 6 + 5
•
1902 1 + 9 + 02 = . 56743 5 × (67 + 4) + 3
2 + 05 × 1 2051 = . • 63874 6 × 3 × 8 + 74
•
1789 1×7+8+9 = . 25046 (2 + 50 + 4) × 6
19 + 03 1903 = . • 64875 (6 + 4) × (8 + 7) × 5
1792 (1 + 7 × 9) × 2 • = . 63504 (6 + 3) × 504
•
1794 1 + 7 × (9 + 4) = . 20865 (208 + 6) × 5
•
•
1806 1 × 8 + 06 = . 42957 4 + ((2 + 9 × 5) × 7)
•
•
•
•
2106 2 × 10 + 6 = . 34587 (3 + 4) × (5 + 8 × 7)
•
(2 + 10) × 6 2106 = . 43875 (4 × 3 + 8) × 75
•
2106 2 × 10 + 6 = . 54837 5+4×8×3×7
•
2106 2 + 1 + 06 = . 58734 5 + 8 + 7 × 34
•
2109 2+1+0×9 = . 43586 43 + 5 + 8 + 6
•
2109 2+1+0×9 = . 83657 8 × (3 + 6 + 5) + 7
•
2134 2 × (1 + 3) + 4 = . 65087 6 × (5 + 08 × 7)
•
2 × 1 × (4 + 8) 2148 = . 59607 59 + 607
•
2159 (2 + 1) × (5 + 9) = . 73406 7 × 34 × 06
•
2169 2 + 1 + 69 = . 85073 8 × (50 × 7 + 3)
•
(2 + 1 + 7) × 5 2175 = . 96048 (9 + 60) × 4 × 8
2 + 05 × 8 2058 = . 17346 1 + 7 + 346
•
•
1×9+0×8 1908 = . 37524 3 × (7 × 5 + 24)
•
2 × 06 × 4 2064 = . 13975 1 + 3 × 9 × (7 + 5)
1 × 80 + 6 1806 = . 45927 (4 + 5) × 9 × 27
•
1924 1×9×2×4 = . 36075 3 × 6 × 075
•
2064 (2 + 06) × 4 = . 38571 3 + 85 × 7 × 1
•
2176 2 × (1 + 7) + 6 = . 35904 359 + 04
1806 1 × 8 × 06 = . 97524 9 × (7 + 5) × 24
•
1926 1 × (9 + 2) × 6 = . 47508 4 × (7 + 50 × 8)
•
2068 20 + 68 = . 17954 (1 + 7) × 95 + 4
•
2×1×7+8 2178 = . 65043 650 + 4 + 3
•
2187 21 + 87 = . 45603 4 × (560 + 3)
•
2189 21 + 89 = . 70645 (706 + 4) × 5
•
2196 2×1×9×6 = . 35807 3 × (580 + 7)
•
2304 2 × (30 + 4) = . 17856 17 + 85 × 6
•
2304 (2 + 30) × 4 = . 18576 18 × 57 + 6
•
2304 (2 + 30) × 4 = . 71568 7 × 1 × 568
1809 1 + 8 + 09 = . 47235 47 × (2 + 3 + 5)
1932 1×9+3+2 • = . 57408 (5 + 7 + 40) × 8
•
1836 1×8+3+6 = . 52704 (52 + 70) × 4
•
•
1842 1×8×4×2 = . 96705 96 × 7 × 05
•
1×8×4×6 1846 = . 23075 (2 + 30) × 75
•
•
2105 (2 + 1) × 05 = . 37469 3 × (74 + 6 + 9)
1905 1 + 9 + 05 = . 23876 2 + 3 × (8 × 7 + 6)
1809 1 + 8 + 09 • = . 45627 4 × 5 + 62 × 7
•
1904 1 + 9 + 04 = . 52768 52 + 7 × 6 × 8
2051 2 + 05 × 1 • = . 93467 9 × 34 + 6 + 7
2097 2 + 09 + 7 = . 35416 (3 × 5 + 4) × 16
•
2035 20 + 35 = . 17649 (1 + (7 + 6) × 4) × 9
1 + 75 + 2 1752 = . 94608 9 × (460 + 8)
•
•
1952 (1 + 95) × 2 = . 40687 (40 + 6) × 87
2061 2 + 06 + 1 = . 34579 3 × 45 + 7 + 9
2079 2 + 07 + 9 • = . 68145 (6 + 8 × 14) × 5
•
2086 2 × (08 + 6) = . 15347 1 + 5 × (34 + 7) 2093 2 + (09 + 3) = . 65481 6 + 54 × 8 × 1
1953 1×9+5×3 = . 24087 240 + 8 × 7
•
•
1958 1×9+5+8 = . 64703 6 × 4 + 703
•
2095 2×0×9+5 = . 74163 (7 × 4 + 1) × 6 + 3
•
1972 19 + 7 + 2 = . 68034 6 + 80 × 3 × 4
•
2097 2 + 097 = . 16543 1 + 65 × 4 × 3
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
2345 (2 + 3 + 4) × 5 = . 61908 6 × (190 + 8)
•
2479 2 × (4 × 7 + 9) = . 61305 6 × 1 × 305
•
2679 2+6+7+9 = . 40185 4 × 018 × 5
•
2871 2 × 87 × 1 = . 35046 (350 + 4) × 6
•
2345 (2 + 3 + 4) × 5 = . 70819 (70 + 81) × 9
•
2486 2×4+8+6 = . 53901 53 × 9 × 01
•
2691 2+6+9+1 = . 57408 (5 + 7) × 4 × 08
•
2871 2×8+7+1 = . 43065 4 × 3 × 06 × 5
•
2+8+9+3 2893 = . 40765 40 × 7 + 6 × 5
•
2904 (2 + 9) × 04 = . 17358 17 × 3 × 5 + 8
•
2904 2 × (9 + 0 × 4) = . 35816 3 × (58 + 16)
•
2904 (2 + 9) × 04 = . 37158 37 × 15 + 8
2354 (23 + 5) × 4 • = . 91806 91 × 8 × 06
2506 (2 + 5) × 06 • = . 18437 1 × 8 + 43 × 7
2376 23 + 7 + 6 = . 14058 (1 + 40) × 5 + 8
2508 (2 + 5) × 08 • = . 19437 (19 + 43) × 7
2384 (2 + 3) × (8 + 4) = . 75096 7 × 5 × 09 × 6
•
•
2397 2+3+9×7 = . 14805 1 × (4 + 80) × 5
•
2401 2 + 40 × 1 = . 38759 3 + (8 + 7) × 5 × 9
•
•
2403 (2 + 4) × 03 • = . 18957 (18 + 9) × 5 + 7
•
2403 2 + 4 + 03 = . 19758 1 × 9 + 7 + 58
•
2604 2 × 60 + 4 = . 18375 (1 + 8 × 3) × 7 × 5
•
2708 2+7×0×8 = . 96134 9 × 6 + 13 + 4
•
2904 (2 + 9) × 04 = . 51876 (5 + 18 × 7) × 6
•
(2 + 60) × 4 2604 = . 91875 (9 + 1) × 875
•
2 × (7 + 1 + 6) 2716 = . 83905 (83 + 90) × 5
•
2 + 9 + 04 2904 = . 73568 (7 + 3) × (5 × 6 + 8)
•
2608 2 × 6 × 08 = . 34719 (3 × 47 + 1) × 9
2745 (2 + 7) × 4 × 5 • = . 13908 1 + 3 + 908
•
2907 2 + 9 + 07 = . 46835 4 + 6 + 8 × 35
•
2915 2 × (9 + 1 + 5) = . 47806 (4 + 78) × 06
•
2945 (2 + 9 + 4) × 5 = . 31806 3 + 1 + 806
•
2948 2 × (9 × 4 + 8) = . 10653 1 × 06 × 53
2786 2×7+8+6 = . 59103 591 + 03
•
2948 2 × (9 × 4 + 8) = . 13065 13 × 06 × 5
2608 2 + 6 + 08 • = . 43195 (43 + 1 + 9) × 5
•
2415 2 × 4 × 15 = . 36708 3 × (6 + 70) × 8
•
•
2451 2 × (4 + 5) + 1 = . 80367 (80 + 3 + 6) × 7
•
2451 2 + 4 × (5 + 1) = . 90687 906 + 8 × 7
•
2457 (2 × 4 + 5) × 7 = . 13608 1 × (3 + 60) × 8
2704 2 + 7 + 04 = . 81536 8 × (1 + (5 + 3) × 6)
2×7+5×4 2754 = . • 16038 160 + 38
•
2786 2×7+8+6 = . 10945 10 × 9 + 4 × 5
2613 2×6×1+3 = . 90584 (90 + 5 × 8) × 4
•
•
2613 26 + 13 = . 90785 90 × (7 + 8) + 5
•
2 × (7 + 9) + 8 2798 = . 50364 5 × 036 × 4
•
2 × (9 × 4 + 8) 2948 = . 61305 6 × 1 × 305
•
2618 2×6+1+8 = . 35904 (3 + 5) × 9 × 04
•
2805 2 × 80 + 5 = . 17493 1 × 7 × 49 × 3
•
2958 29 × 5 + 8 = . 14036 (1 + 40 × 3) × 6
2634 2 × 6 × (3 + 4) • = . 10975 (1 + 09) × 7 × 5
•
2805 2+8+0×5 = . 43197 (4 × 3 + 1 + 9) × 7
•
2985 (29 + 8) × 5 = . 10746 (107 + 4) × 6
•
3015 3 × 01 × 5 = . 68742 6 × (8 + 7 + 42)
•
3015 3 × 01 × 5 = . 97284 (9 + 7 × 2 × 8) × 4
•
3×0×1+7 3017 = . 68529 6 + 8 + 5 × 29
•
3024 3 × 02 × 4 = . 58716 5 × 8 + 71 × 6
•
2457 (2 + 4) × 5 × 7 = . 30186 30 × 1 × 86
•
•
2473 2 × (4 + 7) + 3 = . 81609 816 + 09
•
•
2479 2 × (4 × 7 + 9) = . 10653 1 × 06 × 53
•
2479 2 × (4 × 7 + 9) = . 13065 13 × 06 × 5
•
2704 2 × 70 + 4 = . 13689 (13 + 68) × 9
•
2 × 40 + 8 2408 = . 65317 (6 + 5) × 31 × 7
2×4+5×3 2453 • = . 90761 90 + 761
•
2 + (5 + 8) × 3 2583 = . 10647 1 + 06 × 4 × 7
(2 + 6 + 1) × 3 2613 = . • 47905 (4 + 7) × 9 × 05
•
2701 2 + 70 + 1 • = . 43956 4 × 3 × 9 × (5 + 6)
2637 2+6×3+7 = . 58014 580 + 14 2638 2+6×3+8 = . 14509 145 + 09
2815 (2 + 8) × (1 + 5) • = . 34906 (34 + 90) × 6
•
2816 (2 + 8) × 16 = . 57904 5 × 7 × (90 + 4) 2831 2×8+3×1 = . 75096 (75 + 09) × 6
2653 2 × (6 + 5 + 3) = . 19708 (19 + 7) × 08
•
•
2673 2×6+7×3 = . 10854 10 × (8 + 5) + 4
•
2834 (2 + 8 + 3) × 4 = . 19075 (1 + 9) × 07 × 5
•
3024 3 × 02 × 4 = . 67158 (6 + 7) × (1 + 5 × 8)
•
2678 2+6×7+8 = . 30591 3 + 0591
•
2835 2 + 8 + 35 = . 10647 1 + 06 × 4 × 7
•
3 × (04 + 2) 3042 = . 76895 (7 × 6 + 8) × 9 + 5
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
3042 3 × (04 + 2) = . 81965 8 × (1 + 9) × 6 + 5
•
3084 30 + 8 + 4 = . 79156 (7 + 91) × (5 + 6)
•
3207 3 × (2 + 07) = . 69485 (69 + 48) × 5
•
3045 3×0×4+5 = . 71862 7 × 1 × 8 + 62
•
3087 3 × (0 × 8 + 7) = . 12495 1 + (2 + 4) × (9 + 5)
•
3046 3×0×4+6 • = . 25891 2+5×8+9×1 3 × (0 × 4 + 8) 3048 = . • 17526 1 + 7 + 5 × 26
•
3048 3 × (0 × 4 + 8) = . 69215 (6 × 9 × 2 + 1) × 5
3087 3 + 08 + 7 • = . 45962 4 × (59 + 6 + 2)
•
•
3096 3 × 096 = . 17458 1 × 7 × 4 × 58
•
3507 35 + 0 × 7 = . 82164 82 × 1 × (6 + 4)
3208 3 × 2 × 08 = . 46917 4 + 691 + 7
•
3507 35 + 07 = . 92184 92 × 1 × (8 + 4)
3208 3 × 2 × 08 • = . 61754 6 + 17 × 54
•
35 + 09 3509 = . 17864 1 × 7 × (8 + 6 × 4)
•
3509 35 + 09 = . 61248 6 × (12 + 4) × 8
•
3514 3 + 5 × (1 + 4) = . 89607 (8 + 9) × 6 × 07
•
3524 (3 + 5) × 2 + 4 = . 60789 6 × 07 × 8 + 9
•
3564 3+5+6+4 = . 17028 1 × 70 + 2 × 8
3216 32 + 16 • = . 90785 90 × (7 + 8) + 5
3096 (3 + 09) × 6 = . 17845 (1 + 78 + 4) × 5
•
3256 32 + 56 = . 18907 (1 + 8 × 9) × 07
3052 3 × (05 + 2) = . 94176 9 × (4 + 1 + 7) × 6
•
3097 3 + 09 + 7 = . 18256 1 × 82 + 5 × 6
•
3258 3 + 25 + 8 = . 10679 1 + (06 + 7) × 9
•
3056 (3 + 05) × 6 = . 19482 1 × 9 × (4 × 8 + 2)
•
3105 3 × (10 + 5) = . 28497 (2 + 8 + 49) × 7
•
3267 3 × (2 + 6 × 7) = . 15048 150 × 4 + 8
•
3 + 5 + 64 3564 = . 71082 (710 + 8) × 2
•
3 + (0 × 5 + 8) 3058 = . 29746 2 + 9 × (7 + 4) + 6
•
3 + 108 3108 = . 59472 59 × 4 × (7 + 2)
•
3 + 2 × (8 + 6) 3286 = . 19504 (1 + 9 × 5) × 04
•
3567 3 × 5 + 67 = . 28014 (2 × 80 + 1) × 4
•
3068 3 × 06 + 8 = . 21594 2+1+5×9×4
3108 3×1+0×8 • = . 94276 9 + 4 + 2 + 76
3289 32 + 8 × 9 • = . 67045 (6 × 70 + 4) × 5
•
3591 3+5+9+1 = . 78204 (78 + 20) × 4
•
3627 3+6+2+7 = . 95108 (9 + 5 × 10) × 8
•
3 × (6 + 2 + 8) 3628 = . 19047 1 × 9 × 04 × 7
•
3629 3 + 6 + 29 = . 48705 (4 × 8 + 70) × 5
•
3648 3×6+4×8 = . 51072 5 × 10 × 7 × 2
•
3068 3 × 06 + 8 • = . 59472 (5 + 9) × 4 × (7 + 2)
•
3069 30 + 69 = . 14725 (1 + 47 × 2) × 5
•
•
3108 3 + 10 + 8 = . 94572 9 + 45 × 7 × 2
•
3402 3 × (40 + 2) = . 15768 1 × 576 + 8
3 + 1 × 27 3127 = . 46905 (4 + 6) × 9 × 05
•
3164 3+1+6×4 = . 80795 (80 + 7 × 9) × 5
•
3402 (3 + 4) × 02 = . 75816 (7 + 5 × (8 + 1)) × 6
3 × (40 + 2) 3402 = . 75168 (7 + 51) × 6 × 8
3071 30 + 7 × 1 = . 25896 2 + 5 × (8 + 9 × 6)
•
•
3071 30 + 7 × 1 = . 46895 (4 × 6 + 89) × 5
•
(3 + 1) × 6 + 8 3168 = . 75042 750 + 4 × 2
•
3406 3 + 4 + 06 = . 15982 1×5×9+8×2
•
3+6+5×4 3654 = . 70812 70 × 8 × 1 + 2
•
3074 30 + 7 × 4 = . 12985 (1 + 29) × 8 + 5
•
3186 3 × (1 + 8) × 6 = . 20945 (209 + 4) × 5
•
3+4+0×9 3409 = . 26785 2+6+7+8×5
•
3682 3×6+8+2 = . 17095 (17 + 09) × 5
•
3702 3+7×0×2 = . 41956 4 + 19 + 5 + 6
•
3704 3 + 7 + 04 = . 98156 (9 × 8 + 1) × 5 + 6
•
3705 3 × (70 + 5) = . 29146 (291 + 4) × 6
•
3705 3 + (7 + 05) = . 46189 (4 + 6 + 1) × (8 + 9)
•
3712 3 + 7 × (1 + 2) = . 96048 9 + 604 + 8
•
3715 (3 + 7 + 1) × 5 = . 46809 4 + 680 + 9
•
3 + 078 3078 = . • 16492 (1 + 6 × 4 × 9) × 2
•
•
•
•
3078 3 + 078 = . 21964 2 + 1 × 9 × 64
3×1×9×5 3195 = . • 67024 (6 + 702) × 4
•
3078 3 + 07 + 8 = . 46512 4 × (65 + 1 + 2)
•
3082 30 + 8 × 2 = . 16549 1 + 6 × (5 + 4 × 9)
•
3 × (0 × 8 + 4) 3084 = . 65792 (65 + 7 × 9) × 2
•
3204 3 × 2 × 04 = . 18957 (18 + 9) × 5 + 7
3451 (3 + 4) × (5 + 1) • = . 76908 (7 + 6) × 9 × 08
•
3478 3 × 47 × 8 = . 15096 (1 + 50) × 96 3489 3 × (4 + 8 + 9) = . 72106 (7 + 210) × 6
3204 3 × (2 + 04) = . 19758 1 + 97 + 5 + 8
•
3204 3 × (20 + 4) = . 86597 (8 + 6 × 5 × 9) × 7
•
3501 3 + 5 + 01 = . 96472 (96 + 4 × 7) × 2
3+2+0×5 3205 = . 48716 4 × (8 + 7) + 16
•
3504 3+5×0×4 = . 68912 6×8+9+1×2
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
3746 3+7+4×6 = . 28095 2 × 80 + 95
•
3927 3 + 92 + 7 = . 46508 4 × 6 × 50 + 8
•
4065 4 + 06 + 5 = . 19783 1 + (9 + 7 + 8) × 3
•
3762 3 × (7 + 6 × 2) = . 14058 (1 + 40) × 5 + 8
•
3941 3+9+4×1 = . 86702 8 × (6 × 7 + 02)
•
4069 4+0×6+9 = . 17528 (1 + 7) × 5 + 2 × 8
3768 3+7+6+8 • = . 21509 2 + 15 × 09
3 + 9 + 42 3942 = . • 10658 106 + 5 × 8
4075 40 + 7 × 5 • = . 31296 3 × 1 × 2 × 96
3781 3+7+8+1 • = . 40596 4 × (05 × 9 + 6)
3952 3 × (9 + 5 + 2) • = . 70148 (70 + 1) × (4 + 8)
4086 40 + 8 + 6 • = . 32915 3 × 29 × 1 × 5
3 × (8 + 0 × 4) 3804 = . 67521 6 × (7 × 5 × 2 + 1)
•
•
3807 3 + 8 + 07 = . 45261 4 × 52 + 6 × 1
•
4 × (08 + 7) 4087 = . 65392 6 + 53 × 9 × 2
•
4208 4 × 2 × 08 = . 79163 7 × (9 + 163)
•
4218 4 + 218 = . 37905 3 × 7 × (90 + 5)
•
4251 4 × (2 + 5 + 1) = . 80769 8 × (07 + 69)
•
(4 + 2 + 5) × 7 4257 = . 30186 30 × 18 + 6
•
4257 42 + 57 = . 31089 3 + 10 × 8 × 9
•
4257 4×2+5×7 = . 38016 3 × 8 × 016
•
4268 (4 + 2) × 6 + 8 = . 13095 1 × 3 × 09 × 5
3968 3 × (9 × 6 + 8) = . 21504 2 × 1 × 504
•
•
3971 3+9+7×1 = . 65208 6 × 52 + 0 × 8
•
4095 40 + 95 = . 17836 1 + 7 × 83 + 6
3807 3 + 8 + 07 = . 65142 (6 + 5) × 14 × 2
•
3971 3+9+7×1 = . 80465 80 × 4 + 65
•
4096 4 × (0 × 9 + 6) = . 15872 1 × (5 + 8) × 7 + 2
•
4275 (4 × 2 + 7) × 5 = . 31806 3 × (180 + 6)
•
3×8+2+9 3829 = . 17504 (1 + 7) × 5 × 04
•
(3 + 9) × (7 + 2) 3972 = . 45016 4 × (50 + 1) × 6
•
4097 4×0×9+7 = . 53261 5 × 3 × 2 + 61
•
4278 4+2+7×8 = . 95013 9 × (50 + 1) × 3
3+8+5×2 3852 = . 71904 (7 + 1 + 90) × 4
•
3976 3×9+7+6 = . 80514 805 + 1 + 4
•
4 + 107 4107 = . 25863 2 × 58 × 6 + 3
•
•
4279 4+2+7+9 = . 63018 6 × 3 × 018
•
3864 3 × (8 + 6) × 4 = . 10925 (10 + 9) × 25
3987 3 + 98 + 7 • = . 15062 (1 + 50) × (6 + 2)
4302 4 + 30 + 2 = . 76958 7 × (6 × (9 + 5) + 8)
•
4305 (4 + 3) × 05 = . 28167 2 × 81 + 67
•
(4 + 3) × 05 4305 = . 72816 72 × 8 + 16
•
4307 4×3+0×7 = . 68912 (6 + 89 + 1) × 2
•
4316 4 × (3 + 1) × 6 = . 80925 8 × 09 × 25
•
4325 (4 + 3 × 2) × 5 = . 79061 7 + 906 + 1
•
43 + 2 + 9 4329 = . 61087 6 + 108 × 7
•
4368 43 + 6 × 8 = . 10752 (107 + 5) × 2
•
4368 43 + 6 × 8 = . 15072 (150 + 7) × 2
•
4371 4 + 3 × 71 = . 26508 2 × (650 + 8)
•
4379 (4 + 3) × 7 + 9 = . 10268 1 × 02 × 68
•
4392 4×3×9×2 = . 10675 (1 + 06) × 75
•
•
•
•
4029 4 × 02 + 9 = . 17538 1 + 7 × 5 + 38
•
4109 4 × 10 + 9 = . 37568 ((3 + 7) × 5 + 6) × 8
3892 3×8+9×2 = . 10564 10 × (5 + 6) + 4
•
4029 40 + 2 + 9 = . 38157 3 × (8 + 15) × 7
•
4125 (4 + 1) × 25 = . 70983 (709 + 8) × 3
3 + 90 + 1 3901 = . 25647 2 + 56 × (4 + 7)
4029 4 × 02 + 9 = . • 81765 ((8 + 1) × 7 + 6) × 5
•
4128 4 + 12 + 8 = . 50396 5 + 03 × 96
•
•
3874 3×8+7×4 = . 12069 (1 + 2) × 06 × 9
•
•
•
3901 3 + 90 + 1 = . 68475 6 × (8 + 47) × 5
•
3904 3 + 9 + 04 = . 76128 (7 + 6) × (1 + 2) × 8
•
3908 3 + 9 + 08 = . 54712 5×4×7×1×2
•
•
•
•
4 + 1 + 08 4108 = . 79632 7 × (9 + 6 + 3) × 2
4 × 016 4016 = . 27359 2 + 7 × (3 + 59)
38 + 6 + 4 3864 = . 12075 1 × 2 × 075
•
4107 4 + 10 × 7 • = . 63825 (6 × 38 + 2) × 5
4032 (4 + 03) × 2 = . 17568 1+7+5+6×8
4163 4+1+6×3 • = . 89052 (8 + 90) × 5 + 2 4168 4 × (16 + 8) = . 39075 (3 + 9) × 075
4032 4 × (0 × 3 + 2) = . 85176 (8 + 5 × 1) × (7 + 6)
•
•
4056 4 × 05 + 6 = . 13728 (1 + 3) × (7 × 2 + 8)
•
3915 3×9×1×5 = . 27608 2 × 7 × (60 + 8)
•
4056 40 + 56 = . 92781 9 + 27 × 81
•
4193 41 + 9 + 3 = . 58702 5 × 8 + 702
3921 3+9×2×1 = . 75806 7 × 58 + 0 × 6
•
4059 40 + 59 = . 16728 ((1 + 6) × 7 + 2) × 8
•
4+1×9+8 4198 = . 50376 (5 + 037) × 6
4176 41 + 7 + 6 = . 39208 3 × (9 + 20 × 8)
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
4503 4 + 50 + 3 = . 26781 2+6×7×8+1
•
4752 4 × (7 + 5 + 2) = . 16038 1 + 60 × 3 + 8
•
5048 5 × 04 + 8 = . 73196 7 × (3 + 1 + 9 × 6)
•
4509 45 + 09 = . 17368 (1 + 7 + 3 × 6) × 8
•
4763 4+7×6×3 = . 19052 (1 + 9) × 052
•
5049 (5 + 04) × 9 = . 18326 (1 + 8 × 3 × 2) × 6
4512 4 × (5 + 1) × 2 • = . 83096 830 + 9 × 6
•
4+5+2×1 4521 = . 80967 80 + 9 × (6 + 7)
4792 4 × (7 + 9 × 2) • = . 50316 50 × 3 × (1 + 6)
•
•
5149 5 × (1 + 4) + 9 = . 72086 7 × (20 + 8 × 6)
•
5192 5+1×9+2 = . 80476 80 + 4 × 7 × 6
•
5194 5+1+9×4 = . 27083 27 × 08 + 3
•
5203 5 + 2 × 03 = . 48719 (4 + 8) × 7 + 19
•
5208 5 × (20 + 8) = . 13764 1 × 37 × (6 + 4)
•
5208 5 × 2 × 08 = . 14973 (14 + 9) × (7 + 3)
•
5236 5 + 2 × 36 = . 17408 (1 + 7) × 4 × 08
•
5236 5×2+3×6 = . 19074 1 + 90 + 7 + 4
•
5 × 2 × (4 + 8) 5248 = . 30176 30 × (17 + 6)
•
5291 5 × (2 + 9 × 1) = . 30784 (3 + 07) × 8 × 4
•
5304 (5 + 3) × 04 = . 21879 2 × (1 + 8 × 7 + 9)
•
5314 5+3+1×4 = . 69082 (6 + 9 × 08) × 2
•
5382 (5 × 3 + 8) × 2 = . 71604 7 + 1 + 604
5082 50 + 8 × 2 = . 74613 (7 × 46 + 1) × 3
•
5391 5+3+9+1 = . 70682 (70 + 6 × 8) × 2
50 + 49 5049 = . • 37281 3 + 728 × 1
48 + 06 4806 = . 13795 (1 + 3 × 7 + 9) × 5
•
4816 4 × (8 + 16) = . 52073 5 × 207 + 3
•
5061 5 + 06 + 1 = . 48923 4 + 8 × (9 + 2 + 3)
4581 4 + 58 + 1 = . 39702 39 × 7 × 02
•
•
4 + 6 + 02 4602 = . 51389 5 × (1 + 3 × 8) + 9
•
4872 4 × (8 + 7) × 2 = . 61509 6 + 1509
•
5069 5 + 069 = . 21783 21 × (7 + 8) + 3
•
4 × 6 + 08 4608 = . 53712 53 × 7 × 1 + 2
•
4896 48 + 9 × 6 = . 10752 (107 + 5) × 2
•
5072 50 + 7 × 2 = . 61498 (61 + 4 × 9) × 8
•
4608 4×6+0×8 = . 97152 9 + 71 × (5 + 2)
•
4896 48 + 9 × 6 = . 15072 (150 + 7) × 2
•
5076 5 + 07 × 6 = . 13824 13 × 8 + 24
4 × 9 + 02 4902 = . 18576 1 × 8 × (5 + 7 + 6)
5 + 076 5076 = . • 93248 93 × (2 × 4 + 8)
4905 4 × 9 × 05 = . 23871 2 + 3 + 871
5082 50 + 8 × 2 • = . 37961 3 + 7 × (9 + 61)
•
4609 4 × 6 + 09 • = . 87152 8 × (71 + 5 + 2) 4617 46 + 1 + 7 • = . 25308 (2 + 5 + 30) × 8
•
•
4623 4 + 62 + 3 = . 90785 90 × (7 + 8) + 5
•
•
4653 4+6×5×3 = . 71082 (710 + 8) × 2
•
•
•
•
•
4972 4+9+7+2 = . 18306 (1 + 8) × (3 + 06)
•
4685 4+6+8×5 = . 30921 30 × (9 + 2 × 1)
•
4981 4 + 98 × 1 = . 30765 3 × 07 × 6 × 5
•
5094 5 + 09 + 4 = . 17263 1 + (7 × 2 + 6) × 3
•
5392 (5 × 3 + 9) × 2 = . 68074 (6 + 80) × 7 + 4
4705 (4 + 7) × 05 = . 16938 1 × 6 × (9 + 3 × 8)
•
4982 (4 + 9) × 8 + 2 = . 10763 1 + 076 × 3
•
5096 (5 + 09) × 6 = . 14378 1 + 4 × (3 + 7 × 8)
•
5406 5 + 40 + 6 = . 72398 (72 + 3) × 9 + 8
•
5421 5+4×2×1 = . 73809 7 × 3 × 8 + 09
•
5423 (5 × 4 + 2) × 3 = . 76908 (7 + 6) × 9 × 08
•
5472 5 × (4 + 72) = . 10368 10 × (3 + 6) × 8
•
5472 5 × (4 + 7) + 2 = . 38016 380 + 16
•
5481 54 + 8 + 1 = . 20793 2 + 079 × 3
•
5496 5 + 4 + 96 = . 38701 3 × 8 × 7 + 01
4731 4 + 73 + 1 = . 26809 26 × (8 + 09)
50 + 1 + 6 5016 • = . 27984 2 + (7 + 9 × 8) × 4
•
4735 4+7×3+5 = . 92806 (9 × 2 + 80) × 6
•
•
4752 (4 × 7 + 5) × 2 = . 10368 1 × 03 × 6 × 8
•
4752 (4 × 7 + 5) × 2 = . 13608 1 + 3 × 60 + 8
•
5082 (5 + 08) × 2 = . 43197 4 × 31 + 97
4953 (4 + 9 + 5) × 3 = . 61087 610 + 8 × 7
4 × 72 × 8 4728 = . • 19503 1 + 9503
•
5 + 06 + 7 5067 = . 21394 2 × (1 + 3) × 9 + 4
5019 5 + 01 × 9 = . 24378 (2 + 4) × (3 + 7) + 8
5 + 10 × 3 5103 • = . 24786 2 × (4 + 78) + 6
•
5103 51 + 03 = . 46872 4 × (6 + 8 × 7) × 2 5103 5 + 103 = . 62748 (6 × 27 + 4) × 8
5019 5 + 01 × 9 = . 37284 (3 + 7 + 2 × 8) × 4
•
•
5046 5 + 04 × 6 = . 18792 1 + 8 + 7 + 92
•
5129 5 × 12 + 9 = . 43708 4 + (3 + 70) × 8
•
5046 5×0×4+6 = . 78213 (7 + 8 × (2 + 1)) × 3
•
5 × 14 + 8 5148 = . 32076 3 × (20 + 7) × 6
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
5632 (5 + 6) × 3 × 2 = . 17408 17 × (4 + 08)
•
6024 60 × (2 + 4) = . 57981 5 × 7 × (98 + 1)
•
6147 6+1+4+7 = . 23905 (2 + 3 + 9) × 05
•
6372 6 × (3 + 7) × 2 = . 18054 (1 × 80 + 5) × 4
•
5643 (5 + 6) × 4 × 3 = . 27018 2 + 70 × (1 + 8)
•
6025 6 × 025 = . 13978 (1 + 3) × (9 + 78)
•
6194 6+1×9+4 = . 20538 2 + 053 + 8
•
6372 6×3+7+2 = . 85904 8 × 5 × 9 + 04
5649 5 + 6 + 49 = . 20713 20 × (7 + 1 + 3)
•
6027 6 × 02 × 7 = . 98154 9 × 8 × (15 + 4)
•
(6 + 1 + 9) × 5 6195 = . 47082 4 × (70 + 82)
•
•
6381 6 + 38 + 1 = . 90752 90 × 7 + 5 × 2
6201 6 × (20 + 1) • = . 37895 3 + 7 + 8 × 95
•
6384 63 + 8 × 4 = . 10752 10 + 75 × 2
•
6 + 401 6401 = . 38925 (3 + 8) × 9 × 25
•
6402 64 + 02 = . 35987 35 × 9 + 8 × 7
•
6 + 4 + 05 6405 = . 32879 3×2+8+7×9
•
6408 6 × (4 + 08) = . 13795 (1 + 3 × 7 + 9) × 5
•
6409 6 × (4 + 09) = . 15283 (1 + 5) × (28 + 3)
•
6432 64 + 32 = . 90785 90 × (7 + 8) + 5
•
6492 (6 × 4 + 9) × 2 = . 53018 530 + 1 + 8
•
6497 6+4+9×7 = . 10235 1 × 023 × 5
•
6+4+9×7 6497 = . 21805 (2 + 1) × 80 + 5
•
6497 6+4+9×7 = . 28035 280 + 35
(62 + 4) × 8 6248 = . 30175 30 × 17 × 5
•
6501 65 + 01 = . 34278 3 × (4 + 2 × 7 × 8)
6248 6+2+4×8 = . 57013 5 × (70 + 1 × 3)
•
6×5×3+2 6532 = . 70148 70 × 14 + 8
•
•
•
•
5×6×9+1 5691 = . 47082 4 × 70 × 8 + 2 5698 5+6+9+8 = . 13024 1 × 30 × 2 + 4 5704 5 + 7 + 04 = . 26381 2 + 6 × (3 + 8 + 1) 5706 5+7+0×6 = . 29481 29 + 4 × 8 + 1
(5 + 7) × (2 + 4) 5724 = . • 18603 (18 + 60) × 3
•
•
•
•
6 + 02 × 8 6028 = . 13974 1 + 39 + 7 + 4 6039 60 + 39 = . 71248 (71 × 2 + 4) × 8 6039 6×0×3+9 = . 85217 8 × 5 × (2 + 1) + 7 6+0×4+1 6041 = . 89752 (8 + 9 + 7 × 5) × 2
•
6215 (6 + 2) × 15 • = . 78309 7 × 8 × 3 × 09 6231 6 × (2 + 3) + 1 • = . 57084 5 × 7 × 08 + 4 •
•
6048 6 × (0 × 4 + 8) = . 15372 (1 + 53 + 7) × 2 •
•
5724 5 + 7 + 24 = . 80136 (80 + 1 + 3) × 6
•
6048 6 × 04 × 8 = . 17325 (1 + 7 × 3) × 25 •
•
5728 (5 + 7) × 2 + 8 = . 13604 (1 + 3 × 6) × 04
•
6048 6 × 04 + 8 = . 21735 (2 + 1 × 7 × 3) × 5 •
•
•
5 × 7 + 42 5742 = . 36018 3 + 60 × 1 × 8 5819 5+8×1×9 = . 46023 4 + 602 + 3
5824 (5 + 8) × 24 • = . 10976 (1 + 097) × 6
•
•
5907 59 + 07 = . 13246 1 + 3 + 24 × 6 5916 5 + 91 + 6 = . 28304 2 × (8 × 30 + 4)
6 × 048 6048 = . • 27951 2 × 7 × 95 + 1 6048 6+0×4+8 • = . 37152 (3 + (7 + 1) × 5) × 2
•
•
•
6048 (6 × 04) + 8 = . 52731 5 + 273 + 1 6058 6 × (05 + 8) = . 12349 12 + 3 × 49
6 × 2 + 01 6201 = . 83475 (8 × 3 + 4 + 7) × 5
•
•
•
•
6231 62 + 31 = . 90785 90 × (7 + 8) + 5 6×2+3×7 6237 = . 10584 (1 + 05 + 8) × 4 (6 + 2 + 3) × 7 6237 = . 10854 10 × (8 + 5) + 4 6237 6×2×3×7 = . 15048 150 × 4 + 8 6237 6×2+3+7 = . 80514 80 + 51 × 4 6238 6 + 2 × (3 + 8) = . 90451 9 × 045 + 1
6071 6 + 07 × 1 = . 38294 3 × (8 + 2 × 9) + 4
•
6259 6+2+5+9 = . 13087 1 + 3 × (08 + 7)
•
6534 (6 + 5) × 3 × 4 = . 71082 (710 + 8) × 2
60 + 72 6072 = . 13984 (1 + 3 + 9 × 8) × 4
•
6279 6 × (2 + 7) + 9 = . 80431 804 + 3 × 1
•
6574 (6 × (5 + 7)) + 4 = . 19203 19 + 203
5916 5×9×1+6 = . 40832 40 × 8 + 32
•
5934 (5 + 9) × (3 + 4) = . 71208 7 × (1 + 20) × 8
•
6102 6 + 102 = . 38759 3 + 8 + 75 × 9
•
6308 (6 + 3) × 08 = . 42579 (42 + 5 + 7) × 9
•
6574 (6 × 57) + 4 = . 20938 (20 + 9) × 38
•
5976 5 × (9 + 7) × 6 = . 20418 (204 + 1) × 8
•
6138 6+1+3×8 = . 57024 (5 + 7) × 024
•
6309 6 × (3 + 09) = . 51874 518 + 74
•
6579 6 + 5 + 79 = . 10234 10 × 2 × (3 + 4)
•
60 + 1 × 8 6018 = . 27435 2 × (7 × 4 + 3) × 5
•
6145 (6 + 1 + 4) × 5 = . 20893 20 × 8 + 9 × 3
•
6318 6×3×1+8 = . 24057 2 + 40 + 57
•
6593 6+5+9×3 = . 28107 2 × 81 + 0 × 7
•
•
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
6705 (6 + 7) × 05 = . 24138 (2 + 4) × (1 + 38)
•
7029 70 + 29 = . 54386 5 × 4 × 38 + 6
•
7095 70 + 95 = . 12384 (1 + 23) × (8 + 4)
•
6708 6 × (70 + 8) = . 12943 129 × (4 + 3)
•
7032 7 + 03 + 2 = . 69148 6 + (9 + 1 + 4) × 8
•
7098 70 + 98 = . 14365 (1 + 4) × (3 + 65)
6725 (6 + 7 + 2) × 5 • = . 90384 (9 + 03) × 84
•
6 × (7 + 3 + 4) 6734 = . 10582 10 × (5 + 8) + 2
7 × 03 × 5 7035 = . • 18492 184 + 92
•
7035 7 × 03 × 5 = . 29681 2 + 9 × (6 × 8 + 1)
7104 (7 + 1) × 04 • = . 35298 3×5+2×9×8
•
•
7316 7 + (3 + 1) × 6 = . 85904 8 × 5 × 9 + 04
•
7325 (7 + 3) × 2 × 5 = . 16408 (1 + 6) × 4 × 08
•
7395 (7 + 3 × 9) × 5 = . 28014 (2 × 80 + 1) × 4
•
7398 7 + 3 + 98 = . 12604 (1 + 2) × 60 + 4
•
7+4+0×3 7403 = . 65281 6 + 5 × 2 + 81
•
7403 7+4+0×3 = . 91528 (9 + 1 + 5 + 2) × 8
7104 7 × 10 + 4 = . 52896 5 + (2 + 89) × 6
7035 7 + 03 + 5 = . 69412 (69 + 4 + 1) × 2
•
•
7042 7 × (04 + 2) = . 56839 5 × 6 × (8 + 3) + 9
•
71 + 28 7128 = . 34056 (3 + 40) × (5 + 6)
•
(7 + 4) × 25 7425 = . 13608 1 × (3 + 60) × 8
6739 6 × (7 + 3) + 9 = . 58014 580 + 14
•
7042 7 × (0 × 4 + 2) = . 65893 6 × 5 + 8 + 93
•
7128 (7 + 1 × 2) × 8 = . 65043 650 + 4 + 3
•
7425 (7 + 4 × 2) × 5 = . 38016 3 × 8 × 016
•
(67 + 4) × 5 6745 = . 20938 (20 + 9) × 38
•
70 + 4 × 9 7049 = . 83125 (8 × 31 + 2) × 5
•
7182 7 × 18 × 2 = . 39045 (3 × 90 + 4) × 5
•
7436 (7 + 4) × 36 = . 10985 (109 + 8) × 5
•
6801 6 × 8 × 01 = . 79345 7 × (9 + 3 + 4) × 5
7051 70 + 51 • = . 29486 2 + 9 × 4 × (8 + 6)
7182 7 × 1 × (8 + 2) • = . 43605 (4 + 3) × 60 + 5
•
7+4+5+2 7452 = . 80316 8 + 031 × 6
•
7463 7+4×6+3 = . 25901 2 × 59 × 01
•
7483 7+4×8+3 = . 65209 6 × (52 + 09)
•
7486 7×4+8×6 = . 19503 195 + 03
6735 (6 + 7 × 3) × 5 = . 18409 1 + 8 + 40 × 9
•
•
6739 6 × (7 + 39) = . 10548 1 × 054 × 8
•
•
6804 6 × (8 + 04) • = . 25137 (25 + 13) × 7
•
6+8×3+2 6832 = . 49105 (4 × 9 + 10) × 5
•
•
6834 6+8×3+4 = . 21507 2 × 1 × 50 + 7
•
7056 (7 + 05) × 6 = . 12348 (1 + 2) × (34 + 8)
•
7105 7 × 1 × 05 = . 29638 2 + (9 + 6 + 3) × 8
7195 (71 + 9) × 5 = . 60438 60 × (4 + 3) × 8 (7 + 2) × 09 7209 = . 15486 (1 + 5 × 4 + 8) × 6
7056 7 × 056 = . 13824 (1 + 3) × 8 × 24
•
•
7056 7 × 056 = . 13842 1 + 384 × 2
•
7 × (2 + 4) × 5 7245 = . 18906 1 × 8 + 90 × 6
•
7491 74 + 91 = . 23608 (2 + 3 + 60) × 8
6×8+7+3 6873 = . 20145 (20 + 14) × 5
•
7056 7 × 056 = . 18432 1 × 8 × 4 × 32
•
7248 (7 + 2 × 4) × 8 = . 91506 9 + 1506
•
7518 7×5×1×8 = . 20943 20 × (9 + 4) × 3
•
6903 6 + 9 + 03 = . 42185 (4 + 2 × (1 + 8)) × 5
•
(7 + 05) × 6 7056 = . 23814 238 + 1 + 4
•
7298 72 × 9 + 8 = . 36045 360 × (4 + 5)
•
7524 (7 + 5) × (2 + 4) = . 16093 1 + 60 + 93
•
6 × (9 + 2 × 1) 6921 = . 80745 (80 + 74) × 5
7056 (7 + 05) × 6 • = . 43218 432 + 1 + 8
72 × 9 + 8 7298 = . • 50463 504 × (6 + 3)
•
753 (7 + 5) × 3 = . 204816 204 × 8 × 1 × 6
•
7536 (7 + 5 + 3) × 6 = . 40192 40 × (1 + 9 + 2)
•
7546 75 + 4 × 6 = . 19208 1 × 9 × (20 + 8)
•
7548 7+5+4+8 = . 10693 1 + 06 + 9 × 3
•
7596 (7 + 5) × (9 + 6) = . 13082 1 × 308 + 2
•
7596 (7 + 5) × (9 + 6) = . 81024 8 × 10 × 24
•
•
•
6972 69 + 7 × 2 = . 51408 51 × (4 + 08) 7014 7 × 01 × 4 = . 92685 ((9 + 2) × 6 + 8) × 5
•
•
7062 70 + 62 = . 15943 1 + (5 + 94) × 3
•
7068 7×0×6+8 = . 51243 5 × (1 + 2) + 43
•
7301 7 × (3 + 01) = . 45892 4 × (5 + 8 + 9) × 2 7308 7 × 3 + 08 = . 15624 1 × 56 + 2 + 4
•
7028 7 × 028 = . 61495 (6 + 1) × 49 × 5
•
7083 7×0×8+3 = . 51942 5+1×9+4×2
•
7308 7 + 3 + 08 = . 49126 4 + 9 × (1 + 2 × 6)
•
70 + 29 7029 = . 43168 4 × (3 + 16) × 8
•
7092 7 + 09 + 2 = . 16548 1×6×5+4+8
•
7 × 3 + 08 7308 = . 52416 (5 + 2 × 4) × 16
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
7598 7+5+9+8 = . 20436 (20 + 4) × 3 + 6
•
8016 8 × 01 × 6 = . 39245 (3 + (9 + 2) × 4) × 5
•
8103 8 + 103 = . 47596 4 + (7 + 5) × 9 × 6
•
8507 8 × 5 + 07 = . 36924 3 × (6 + 9 + 2) × 4
•
7605 (7 + 6) × 05 = . 31824 31 × 8 + 24
•
8016 8 × 01 × 6 = . 79325 (7 + (9 + 3)) × 25
•
8103 8 + 103 = . 67452 6 × 7 × (4 × 5 + 2)
•
8514 8 × (51 + 4) = . 23607 2 × (3 + 607)
•
7605 (7 + 6) × 05 = . 34281 3 × 4 + 281
8029 80 + 2 + 9 • = . 13764 1 × 3 × (7 + 6) × 4
8×1+0×4 8104 = . • 96235 (9 + 6) × 2 × 3 + 5
•
8514 8 + 5 × (1 + 4) = . 60372 6 × (037 + 2)
•
(7 + 6) × 05 7605 = . 81432 8 × (1 + 43 × 2)
80 × 3 + 6 8036 = . • 15974 1 × 5 × 97 + 4
8126 8 + 1 × 26 • = . 37045 (3 + 7 × 04) × 5
•
85 + 17 8517 = . 23046 2 × 3 × 046
•
8517 85 + 17 = . 32064 3 × 2 × 064
•
8517 85 + 17 = . 46092 4 × (60 + 9) × 2
•
8602 (8 + 60) × 2 = . 19734 (19 + 7) × 3 × 4
•
8613 8 + (6 + 1) × 3 = . 24057 24 + 057
7623 7 × (6 + 2 + 3) • = . 15048 (15 + 04) × 8 7623 7 × (6 + 2 + 3) • = . 48015 480 + 1 × 5 7651 7 × (6 + 5 + 1) • = . 48092 48 × (09 + 2)
•
•
•
•
•
•
•
7 + 80 × 1 7801 = . 32549 3+2×5×4×9
•
•
7851 78 + 51 • = . 20936 20 + 9 × 36
8163 8+1+6×3 = . 50792 (5 + 079) × 2
•
8056 8 + 056 = . 47329 4 × (7 + 3 × 29)
•
(8 + 1 × 7) × 5 8175 = . 64092 6 × (40 + 9) × 2
•
8+6+1+5 8615 = . 93042 9 × 3 × 04 × 2
•
8057 8 × (0 × 5 + 7) = . 14963 1 + 4 + 96 + 3
•
8176 8 × (1 + 7 + 6) = . 24309 24 + 309
•
8624 8 × (6 + 2 × 4) = . 39501 3 + 9 + 501
•
8063 8 + (0 × 6 + 3) = . 17592 1+7+5+9+2
•
8217 (82 + 1) × 7 = . 34056 (3 + 40) × 56
•
(86 + 5) × 2 8652 = . 79104 (7 + 9) × 104
•
8064 8 + 06 × 4 = . 15372 1 + 5 × (3 + 7 + 2)
•
8261 8 × 2 + 61 = . 70594 7 × (0 × 5 + 94)
•
8704 8 × 7 × 04 = . 16592 (1 + 6) × (59 + 2)
•
(8 + 7) × (2 + 4) 8724 = . 93056 930 + 5 × 6
•
8+7+2×5 8725 = . 14309 14 + 3 × 09
•
8726 87 × 2 + 6 = . 30541 30 × (5 × 4 + 1)
•
8732 (8 × 7 + 3) × 2 = . 15096 150 + 9 × 6
•
8742 (8 + 7) × 4 + 2 = . 19035 1 × 9 × 03 × 5
7803 78 + 03 = . 46529 465 + 2 × 9
7832 7 × 8 + 32 = . 19046 190 + 4 × 6
•
8145 8 × (1 + 4) + 5 = . 60273 60 + 273
•
7803 78 + 03 = . 21964 (2 + 1 + 9 × 6) × 4
7×8×1×2 7812 = . 63054 6 × 30 × 5 + 4
8047 8 × 04 + 7 = . 95326 (9 × 5 + 32) × 6
•
80 + 52 8052 = . 69174 6 × 9 × (17 + 4)
•
7803 78 + 03 = . 19652 (1 + 96 + 5) × 2
7805 7 × (80 + 5) = . 12934 1 × 29 × 34
8047 8 × 04 + 7 = . 36521 3 × (6 + 52 + 1)
8172 8 × 1 × (7 + 2) = . 50394 50 + 394
8 + 064 8064 = . • 15792 15 + 7 × 9 × 2
8274 8×2+7+4 • = . 19306 19 × 3 + 06
8064 8 × 064 • = . 23751 2 × (3 + 751)
8304 8 + 30 × 4 • = . 12975 1 + 2 × 97 + 5
•
7854 7 × (8 + 5 + 4) • = . 10296 102 + 9 × 6
•
7912 7×9×1×2 • = . 35604 3 + 560 + 4
•
8064 (8 + 06) × 4 = . 35712 35 × 7 + 1 + 2 8064 (8 + 06) × 4 = . 53712 53 × 7 × 1 + 2
•
•
8316 (8 + 3) × (1 + 6) = . 52704 (52 + 70) × 4 8327 8×3×2+7 = . 59046 5 × (9 + 04) × 6
8075 (8 + 07) × 5 = . 32946 3 × (2 + 94 + 6)
•
8 + (3 + 5) × 2 8352 = . 40716 40 + 71 + 6
•
8742 (8 + 7) × 4 + 2 = . 50619 50 × (6 + 1) + 9
•
8092 (8 + 09) × 2 = . 14756 1 + (4 + 7) × 5 + 6
•
8369 8×3+6+9 = . 25107 2 × 5 + 107
•
8 × (9 + 2) + 4 8924 = . 57036 570 + 3 × 6
•
7938 7+9+3+8 = . 12054 1 + 2 × 05 × 4
•
7945 7+9+4+5 = . 60382 60 × 3 + 8 + 2
•
8096 80 + 96 = . 15732 1 × 57 × 3 × 2
•
8376 8 × (3 + 7) × 6 = . 20591 20 × 59 × 1
•
8926 8 + 92 × 6 = . 35704 (3 + 5) × 70 × 4
•
7952 7 × (9 + 5 + 2) = . 13064 1 × 30 × 6 + 4
•
8103 8 + 103 = . 25769 2 + 57 × 6 + 9
•
8395 (8 + 3 + 9) × 5 = . 47012 4 × 70 × 1 × 2
•
8942 8 + (9 + 4) × 2 = . 60753 6 + 075 × 3
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
8957 8 + (9 + 5) × 7 = . 43602 43 × 6 × 02
•
9078 9+0×7+8 = . 51264 (5 + 1) × (2 × 6 + 4)
•
9306 93 + 06 = . 45872 4 × (5 + 8 × 7) × 2
•
8974 8+9+7+4 = . 10256 1 × 02 + 5 × 6
•
9104 (9 + 1) × 04 = . 35278 3 + (5 + 2 × 7) × 8
•
9324 9 + 32 + 4 = . 58016 5 × 8 × (01 + 6)
8976 8 × (9 + 7 + 6) • = . 20451 20 × 4 × 5 + 1
•
9018 9 + 01 + 8 = . 72645 7 + 2 × (64 + 5)
9126 9 × (1 + 2 + 6) • = . 30758 3 × 07 × (5 + 8)
•
90 + 2 + 4 9024 = . 61758 617 + 5 × 8
•
9025 90 + 2 × 5 = . 31768 (31 + 7 + 6) × 8
•
9128 (9 + 1) × 2 + 8 = . 75306 75 × 3 + 06
•
9572 (9 + 57) × 2 = . 40681 40 × (6 + 8) + 1
•
9632 9 × 6 + 32 = . 51408 51 + 408
•
9648 9+6×4×8 = . 10752 (107 + 5) × 2
•
9648 9+6×4×8 = . 15072 (150 + 7) × 2
•
9 × 6 + 57 9657 = . 28014 2 + 80 × 1 × 4
9+3+6×8 9368 = . 14052 1 × (40 + 5) × 2
•
9671 9 + 6 × (7 + 1) = . 28504 (2 + 8 × 5) × 04
9328 (9 + 3) × 2 + 8 • = . 67045 (6 × 7 + 04) × 5
•
9345 9 × (3 + 4) × 5 = . 18067 (1 + 80 + 6) × 7
9135 9 × 1 × 35 = . 27608 2 × 7 × (60 + 8)
•
•
9135 9×1×3×5 = . 67802 6 × (7 + 80 × 2)
•
9374 9 + 3 + 74 = . 10682 (1 + 06 × 8) × 2
•
(9 + 6 × 7) × 2 9672 = . 14508 145 + 08
9028 9 + 028 = . 31476 3 + (14 + 7) × 6
•
9162 9+1+6+2 = . 73805 (7 × 3 + 8) × 05
•
9 + 3 × (7 + 6) 9376 = . 10548 1 + 05 + 48
•
9681 96 + 8 + 1 = . 20745 20 × (7 + 4) + 5
•
(9 + 03) × 2 9032 = . 84675 (8 × 4 + 6 + 7) × 5
•
91 + 63 9163 = . 82705 (8 + 270) × 5
•
9 × (3 + 7) + 6 9376 = . 58014 580 + 14
•
9684 96 + 8 + 4 = . 20175 (2 + 01) × 75
•
9037 9×0×3+7 = . 65841 6 + 5 + 8 × (4 + 1)
•
9 × (7 + 0 × 2) 9702 = . 18634 1 + 86 + 34
•
9702 97 + 02 = . 31654 3 + 16 × 5 × 4
•
9708 9 + 7 + 08 = . 14562 1×4+5×6+2
•
9+7+2+4 9724 = . 85306 8 + 5 + 30 × 6
•
•
•
9042 9 × 04 × 2 = . 67815 6 × (7 + 8) × (1 + 5)
9164 9 + 16 + 4 • = . 35708 3 × 5 × 7 + 08
•
9418 9 + 41 + 8 • = . 70635 70 × 6 + 3 × 5
9164 9×1×6+4 = . 57038 5 × 70 + 3 + 8
•
9423 9 × (4 + 2 + 3) = . 61075 (6 + 1) × 075
9 × (0 × 4 + 5) 9045 = . 23718 2 × (3 + 7 × 1 × 8)
•
9167 9+1×6+7 = . 82503 (8 × 2 + 50) × 3
•
9438 9 × 4 × (3 + 8) = . 10725 10 × (7 + 2) × 5
•
9045 9 × 045 = . 38726 3 × 8 × 72 + 6
•
9215 (9 × 2 + 1) × 5 = . 30846 308 + 4 + 6
•
9438 (9 + 4) × 3 × 8 = . 12705 12 × 7 × 05
•
9 × (80 + 1) 9801 = . 24563 (24 + 5) × 63
•
9048 9×0×4+8 = . 71253 7 + 1 + 2 + 53
•
9215 (9 × 2 + 1) × 5 = . 36084 360 + 8 + 4
•
9452 (9 + 45) × 2 = . 63801 (6 + 3) × (80 + 1)
•
9801 9 + 801 = . 53724 5 × 37 × 24
•
9048 9 × (0 × 4 + 8) = . 76531 76 × (5 + 3) + 1
9215 (9 × 2 + 1) × 5 = . • 43068 430 + 6 + 8
•
(9 × 8 + 1) × 4 9814 = . 73605 73 × 6 × 05
•
9856 9+8×5+6 = . 17024 1 + 70 + 24
•
9861 9+8×6×1 = . 30275 (3 + 02) × 7 × 5
•
9863 9+8+6×3 = . 50724 5 × (07 + 2) × 4
•
•
9 + 072 9072 = . 81536 8 × (1 + 5 × 3 × 6)
•
9074 9+0×7+4 = . 21638 2 + (1 + 6) × 3 + 8
•
•
9074 9+0×7+4 = . 58632 (5 + 8) × 6 + 3 × 2
•
•
•
9251 (9 + 2) × (5 + 1) = . 40368 4 × (03 + 6) × 8
9472 (9 + 4 × 7) × 2 • = . 10368 10 + 3 + 68
•
9504 95 + 04 = . 31872 318 + 7 × 2 9504 9 × (5 + 04) = . 73216 (7 + 32) × 16
9251 (9 + 2) × 5 × 1 = . 80736 8 × (07 + 3) × 6
•
•
9254 9 × (2 × 5 + 4) = . 31067 3 + 10 × 6 × 7
•
9508 (9 + 5) × 08 = . 26147 261 + 47
•
9864 9 × (8 + 6 + 4) = . 35072 (3 + 5) × 072
9075 9 × 075 = . 12463 1 + 2 × 463
•
9287 92 + 8 × 7 = . 14056 1 × 4 × 056
•
9516 9+5×1×6 = . 27084 27 + 084
•
9873 9 + 8 + 73 = . 50462 5 × 046 × 2
9078 9+0×7+8 = . 41652 (4 + (1 + 6) × 5) × 2
•
9306 93 + 06 = . 14852 148 + 5 × 2
•
9568 (9 + 5 × 6) × 8 = . 17204 1 + 7 × 20 × 4
•
1 + 02 102 = . 389674 3 + (8 + 9) × 674
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
102 1 × 02 = . 439875 (43 + 9 × 8) × 75
•
109 1+0×9 102 1+0×2 = . = . • 872436 87 × (2 × 43 + 6) 498576 4 × (9 + 85) × (7 + 6)
1 + 02 102 = . • 539648 (5 × 396 + 4) × 8
•
1×2+6 126 = . • 380457 3 × (8045 + 7)
102 1 × 02 = . 647598 6 × 47 × 5 × 9 + 8
•
102 1 × 02 = . 748935 (7 + 4) × 89 × 3 × 5
•
•
102 1 × 02 = . 948753 9 × (4 × 8 + 7) × 53
•
•
•
•
143 1+4×3 = . 769208 76 × 920 + 8
•
209 2+0×9 = . 481536 48 × (1 + 5 × 3) × 6
•
241 24 + 1 = . 350896 350 × (8 + 96)
243 (2 + 4) × 3 • = . 106785 10 × (6 + 785)
•
243 2×4×3 = . 170586 (1 + 70 × 5) × 8 × 6
•
305 3 × 05 = . 948672 9 × (4 + 8) × 6 × 72
•
308 3 + 08 = . 127596 1 + 2 + 759 × 6
•
308 3 × 08 = . 149765 (1 + 4 × 97) × 6 × 5
•
3+0×9 309 = . 287164 2 × 87 × 16 + 4
•
309 3 + 09 = . 421785 42 × 1 × 78 × 5
147 1×4+7 = . 259308 (2 + 5) × 9 × 308
•
2+4+6 246 = . 105739 1 + 0573 × 9
•
309 3+0×9 = . 728416 (7 + 2 + 8) × 416
•
152 1+5+2 = . 349068 3 × (4 + 90 × 68)
•
248 2 × (4 + 8) = . 130975 13 × 0975
•
317 3 × (1 + 7) = . 408296 (40 × 8 + 2) × 96
103 1 + 03 = . 427965 (42 × 79 + 6) × 5
•
172 1×7+2 = . 693504 6 × (9 + 3) × 504
•
2 × (4 + 8) 248 = . 361057 (3 + 610) × 57
•
3×2×4 324 = . 106785 10 × (6 + 785)
103 1+0×3 = . 479568 (4 + 79) × 56 + 8
•
176 1×7+6 = . 529408 52 × 94 × 08
•
248 (2 + 4) × 8 = . 671305 6 × 71 × 305
•
324 3 × (2 + 4) = . 107856 1 × 07 × 856
•
326 3 × (2 + 6) = . 418095 4 × (1 + 80) × 95
•
327 (3 + 2) × 7 = . 105948 105 × 9 × (4 + 8)
•
329 3+2+9 = . 540876 (540 + 8) × 7 × 6
•
341 3+4×1 = . 250976 2 + 50 × (97 + 6)
•
342 3+4+2 = . 169708 (1 + 6) × (9 × 70 + 8)
1+0×3 103 = . • 952647 9 + 5 × 264 × 7 •
109 1+0×9 = . 572468 57 × 2 × 46 + 8
104 1+0×4 = . 356928 3 × (5 + 69 × 2) × 8
192 1+9+2 • = . 786304 78 × 630 + 4
•
249 2+4+9 • = . 107568 10 × (75 + 6) × 8
2+0×3 203 2×5+8 258 = .• = . 148596 1 × 4 × (8 × 5 × 9 + 6) 307149 30 × 714 + 9 261 2+6+1 = . 453908 4 × (5 + 3908)
104 1 × 04 = . 387296 38 × 7 × (2 + 9 × 6)
•
•
105 1 × 05 = . 469728 4 × (697 + 2) × 8
•
2+6+4 264 2+0×3 203 = . = . • 159038 1 × 5 + 903 × 8 891576 8 × (91 × (5 + 7) + 6)
•
105 1 × 05 = . 689472 6 × 8 × 9 × (4 + 72)
•
204 2+0×4 = . 615978 61 + 5978
•
273 27 + 3 = . 149058 14 × 90 × (5 + 8)
•
349 3 × (4 + 9) = . 150768 (1 + 50 × 7) × 6 × 8
•
1 × 07 107 = . 582936 5 × 82 × 93 + 6
•
2+0×4 204 = . 716958 71 + 6958
•
291 2+9×1 = . 856704 (8 × 5 + 6) × 704
•
351 3+5×1 = . 247806 2 + 4 + 7 × 806
•
108 1 × 08 • = . 234765 (2 + 3476) × 5
•
•
203 2 × 03 = . 158746 (15 + 87) × 46
205 2 × 05 • = . 316479 31 × 6 × (4 + 79)
108 1+0×8 206 2 + 06 = .• = . 273456 (2 + 7 × 3 × 4 × 5) × 6 457938 (45 + 7) × 9 × 38 108 1+0×8 207 2 + 07 = . • = . 365472 (3 + (6 + 5) × 4) × 72 146395 1 × (4 + 63) × 95
•
357 • 302 3 × 02 140896 • = . 184975 (1 + 8 × (4 + 9)) × 7 × 5 359 • 127086 302 3+0×2 • = . 461758 4 + 61 × 75 + 8 364 • 891072 302 3 × 02 • = . 841976 8 × 41 × (9 + 7 × 6) 376 • 189504
•
108 1+0×8 = . 436752 4 × 3 × (67 × 5 + 2)
•
207 2 × 07 = . 365148 (3 + 6 × 514) × 8
•
304 3+0×4 = . 561792 56 × 1 × (7 + 92)
•
1+0×8 108 = . 743256 74 × 3 × (25 + 6)
•
208 2+0×8 = . 419536 4 × 19 × 53 + 6
•
3 × 05 305 = . 217648 (217 + 6) × 48
=
3 × (5 + 7) . (140 + 8) × 96
=
3×5+9 . 1 × 2 × 708 × 6
=
3+6×4 . (8 + 910) × 72
=
3×7+6 . (18 + 9) × 504
•
381 3+8+1 = . 207645 20 × (7 + 64 × 5)
•
401 4 × 01 = . 385762 38 + 5 × 762
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
402 40 + 2 = . 137685 1 × 3 × 7 × 685
•
436 4 + 36 = . 109872 10 × 9 × 8 × 7 × 2
•
527 5×2+7 = . 693408 6 × (93 × 40 + 8)
•
402 40 + 2 = . 173865 1 × 7 × 3 × 865
•
437 4×3+7 = . 290168 2 + 901 × (6 + 8)
•
•
4 + 02 402 = . 197583 (1 × 975 + 8) × 3
•
4×7×1 471 = . 608532 (60 + 8) × 532
•
403 4 + 03 479 4+7+9 • = •. = . 198276 ((1 + 9) × 8 + 2) × 7 × 6 523068 52 × 30 × (6 + 8) 403 4 × 03 • = . 916825 91 × 6 × (8 + 2) × 5
•
405 4 × 05 = . 371628 37 × 1 × 62 × 8
•
•
4+9+8 498 = . 205176 (205 + 1) × 7 × 6
•
•
•
621 6 + 21 = . 870435 87 × 0435
534 5+3+4 = . 271806 2 + 71 × (80 + 6)
•
624 6×2+4 = . 107835 (1 + 078) × 35
5+3+7 537 = . 842016 8 × 420 × (1 + 6)
•
624 6 × 24 = . 130975 (1 + 30) × 975
•
631 6+3×1 = . 258079 (2 + 5 × 80 + 7) × 9
•
632 (6 + 3) × 2 = . 179804 1 + (7 + 9) × 80 × 4
•
635 (6 + 3) × 5 = . 109728 1 × 0972 × 8
•
639 (6 + 3) × 9 = . 485072 (4 + 850) × 72
562 5 × (6 + 2) = . 301794 30 × 179 × 4
•
643 (6 + 4) × 3 = . 208975 (2 + 08) × 975
539 5 + 39 = . 240786 (2 + 40) × (78 × 6) 541 5×4×1 = . 209367 20 × 9 × (36 + 7)
501 5 + 01 5 × (4 + 3) 543 = . 392784 (3 + 9 × 2) × 7 × 8 × 4 • 278016 = 2 × 7 × 80 × 16 .
406 4+0×6 5 + 01 501 = . 213759 (2 + 1) × (3 + 75) × 9 • 486972 = (4 + 8 + 69) × 72 .
•
4+0×7 407 = . 253968 (25 + 3 × 9) × 6 × 8
•
503 5 + 03 = . 479862 (4 + 79 × 8) × 6 × 2
•
567 5+6+7 = . 103824 (1 + 03) × 824
•
652 6+5+2 = . 149308 1 + 4 × 93 × 08
•
407 4 + 07 = . 286935 (2 × 86 × 9 + 3) × 5
•
5 + 06 506 = . 312984 3 × (1 + 2) × 9 × 84
•
574 5+7+4 = . 196308 19 × (6 + 30) × 8
•
654 6 + 54 = . 109872 10 × 9 × 8 × 7 × 2
•
408 4+0×8 = . 597312 (5 × 97 + 3) × 12
•
506 5 + 06 = . 784392 7 × 84 × (3 × 9 + 2)
•
581 5+8+1 = . 679023 6 × ((7 + 902) × 3)
•
694 6+9×4 = . 375801 3 × (7580 + 1)
•
4+0×9 409 = . 156238 1 × (5 + 62 × 3) × 8
•
507 5 + 07 = . 168324 1 × 6 × 83 × 2 × 4
•
5+9+7 597 = . 142086 (1 + 4 × 208) × 6
•
702 7+0×2 = . 135486 1 + 3 × 5 × (4 + 86)
•
413 4+1×3 = . 697085 (69 + 70) × 85
•
5+0×7 507 = . 241839 (241 + 8 × 3) × 9
602 6+0×2 • = . 318759 (318 + 7 × 5) × 9
•
7 + 02 702 = . 531648 (53 × 16 + 4) × 8
•
703 7 × 03 = . 198246 (1 + 982 + 4) × 6
•
704 7 × 04 = . 356928 (3 + 56 × 9) × 28
•
708 7+0×8 = . 532416 (5 + 324) × 16
•
712 7+1×2 = . 360984 3 + 60 × (9 × 8 + 4)
•
7+1+2 712 = . 563904 5 × (6 + 390) × 4
•
•
415 4+1+5 • = . 679023 6 × (7 + 902) × 3
•
•
415 4×1×5 = . 890673 (8 + 90) × 6 × 73 418 (4 + 1) × 8 = . 305976 305 × (9 + 7) × 6
507 5 + 07 • = . 392418 3 × 9 × (2 + 41) × 8
•
•
423 4+2×3 = . 156087 (1 + 5) × (608 + 7)
•
427 4 × (2 + 7) = . 360815 (3 + 6081) × 5
•
•
429 4×2+9 = . 178035 17 × (80 + 3) × 5
•
432 4×3×2 = . 107856 1 × 07 × 856
•
•
509 5 + 09 = . 216834 21 × (68 + 3) × 4 (5 + 1) × 2 512 = . 369408 3 × (6 + 9 × 40 × 8)
•
•
•
602 6 × 02 = . 354879 (3 + (5 + 4) × 87) × 9 602 6+0×2 = . 413875 (4 + 1) × (3 + 8) × 75 603 6 + 03 = . 428197 42 × 8 × 19 + 7
513 5+1+3 = . 278046 (2 + 7 + 804) × 6
•
516 (5 + 1) × 6 = . 390827 3 × (9082 + 7)
•
604 6+0×4 = . 315892 3 × (1 + 58 × 9) × 2
•
726 7 + 26 = . 159038 1 × 5 + 903 × 8
•
518 5+1×8 = . 307692 30 + 7692
•
607 6+0×7 = . 143859 (1 + 4 × 38 + 5) × 9
•
742 7 × (4 + 2) = . 190853 (1 + 90 × 8 × 5) × 3
•
519 5+1+9 = . 307248 30 × (72 × 4 + 8)
•
608 6 + 08 = . 143792 1 × 43 × 7 × (9 + 2)
•
754 7+5+4 = . 631098 6 × 31 × 09 × 8
603 6 + 03 = . 847952 8 × (4 × 79 × 5 + 2)
Inder J. Taneja
•
756 7+5+6 = . 103824 1 × 03 × 824
•
•
•
984 9+8+4 = . 210576 (2 + 105) × 7 × 6
•
10458 1 × 04 × 5 + 8 = . 27639 2 + 7 × (6 + 3) + 9
762 (7 + 6) × 2 849 8+4+9 = . • = . 140589 (1 + 40) × (5 + 8) × 9 105276 (10 + 52) × 7 × 6
•
10248 10 + 24 + 8 = . 75396 75 + 39 × 6
•
10465 104 + 6 + 5 = . 73892 7 × (3 × 8 + 92)
763 7+6+3 = . 185409 1 × (8 × (54 × 09))
851 8 × (5 + 1) = . 309764 (30 + 9) × 7 × 64
•
10275 1 × 02 × 75 = . 64938 6 + 4 + 938
•
10 + 5 × 27 10527 = . 49368 (4 + 9 × (3 + 6)) × 8
861 8+6×1 • = . 597042 (5 × 970 + 4) × 2
102 + 9 × 6 10296 = . 45738 (4 + 5) × (7 × 3 + 8)
•
•
10527 10 + 5 + 2 × 7 = . 86394 (8 + 6 × 3) × 9 + 4
•
10578 10 × (5 × 7 + 8) = . 32964 (329 + 6) × 4
•
10584 1 + 058 + 4 = . 36792 3 × 67 + 9 × 2
•
10584 1 × 0 × 5 + 84 = . 37296 (3 + 7) × 29 + 6
•
10584 105 + 84 = . 79632 79 × (6 + 3) × 2
782 (7 + 8) × 2 • = . 534106 (5 + 3410) × 6
•
•
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
786 7+8×6 = . 143052 1430 × (5 + 2)
•
•
•
846 8 + 46 = . 109275 (1 + 092) × 75
861 8+6+1 = . 759402 7 × 5 × 9 × (40 + 2)
802 8+0×2 873 8+7+3 = .• = . 456739 4 × (5 + 6 × 7 × 3 × 9) 125906 1 × 2590 + 6
10296 102 + 9 × 6 • = . 45738 (4 + 5) × 7 × (3 + 8)
•
10296 102 + 9 × 6 = . 73458 7 × 3 × (45 + 8) 10368 (1 + 03) × 6 × 8 = . 29754 2 + 9 × (7 + 54)
902 9 + 02 = . 167854 (16 + 7) × (85 + 4)
•
902 9 + 02 = . 345876 (3 × 4 × 58 + 7) × 6
•
1×0×3+6×8 10368 = . 45792 4 × (5 × 7 + 9 × 2)
•
1 + 06 × 4 × 7 10647 = . 53928 (5 × 3 + 92) × 8
904 9 + 04 8 + 03 803 = . = . • 173568 (17 + 35) × 6 × 8 245791 ((2 + 4) × 5 + 7) × 91
•
10368 (1 + 03) × 6 × 8 = . 52974 5 + 2 + 974
•
1 × 0 × 6 + 53 10653 = . 97284 (9 + 7 × 2 × 8) × 4
•
923 9 + 23 803 8+0×3 = . = .• 6 × 784 × 05 927465 (9 + 2) × 7 × 4 × 6 × 5 678405
•
10368 1 × 036 × 8 = . 74592 74 × (5 + 9) × 2
•
10675 1 × 0 × 6 + 75 = . 39284 3 × ((9 + 2) × 8 + 4)
•
80 + 4 804 = . 127635 (1 + 2) × 7 × 635
•
9×2+6 926 = . 137048 (1 + 3 + 70) × 48
•
1 + 03 × 7 + 4 10374 = . 92568 (9 × 2 + 5 + 6) × 8
•
1 × 06 × (9 + 2) 10692 = . 35478 3 + (5 × 4 + 7) × 8
804 8+0×4 • = . 195372 (19 + 5 + 3) × 72
•
952 9+5+2 = . 134708 (1 × 3 + 4 × 70) × 8
•
10395 (10 + 3 + 9) × 5 = . 46872 4 × (6 + 8 × 7) × 2
•
10692 (1 + 06) × (9 + 2) = . 57348 5 × 73 + 48
•
1 + 06 + 9 + 2 10692 = . 75438 7 × (5 + 4 × 3) + 8
•
10695 10 × (6 + 9) + 5 = . 74382 7 × (4 × 38 + 2)
•
1 × 0 × 7 + 25 10725 = . 36894 3 × 6 + (8 + 9) × 4
•
10725 1 × 0 × 7 + 25 = . 39468 3 + 9 + (4 + 6) × 8
•
10738 (10 + 7) × 3 + 8 = . 69524 6 × 9 × (5 + 2) + 4
10437 (10 + 4) × 3 × 7 = . 25986 2 × (5 × 9 × 8 + 6)
•
10764 (1 × 07 + 6) × 4 = . 23598 2 × (3 + 5) + 98
•
10764 10 × (7 × 6 + 4) = . 25389 2 × 538 + 9
•
(1 × 07 + 6) × 4 10764 = . 28359 2 × 8 × (3 + 5) + 9
803 8 + 03 = . 142569 (1 + 42 × 5 + 6) × 9
•
•
803 8 + 03 = . 215496 2 × (1 + 5 × 49) × 6
•
•
•
•
•
•
807 8 × 07 = . 246135 2 × 4 × 61 × 35
953 (9 + 5) × 3 • = . 108642 (108 + 6) × 42
9+5+4 954 809 8+0×9 = . = .• 7 × (1 + 30) × 62 327645 3 × (2 + 7) × 6 × 4 × 5 713062 8+0×9 809 = . 473265 4 × (7 + 32) × 6 × 5
•
•
•
10395 (10 + 3 + 9) × 5 = . 78246 782 + 46 1+0×3+9+5 10395 = . 82467 8 × (2 × 4 + 6) + 7 10395 10 + (3 + 9) × 5 = . 86724 8 × (67 + 2 + 4)
957 9+5×7 = . 128064 (12 + 80) × 64
•
957 9+5×7 = . 812406 812 × (40 + 6)
•
•
961 9+6×1 = . 430528 4 × 30 × (5 + 2) × 8
•
10439 104 + 39 = . 75628 (7 + 5 × 6) × 28
•
9+6×3 963 = . 172805 17 × (280 + 5)
•
1 + 045 × 2 10452 = . 73968 (7 + 39) × (6 + 8)
812 8×1×2 = . 476035 4 × (7 + 60) × 35
•
•
816 (8 + 1) × 6 = . 957032 9 × (5 + 7032)
•
•
827 8+2×7 = . 409365 (40 × 9 + 3) × 6 × 5
•
831 8+3+1 = . 907452 9 × 07 × 4 × 52
•
956 9+5+6 = . 103248 10 × (3 + 24) × 8
10395 (10 + 3 + 9) × 5 • = . 47628 476 + 28
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
10783 1 + 078 + 3 = . 62594 (6 + 2) × 59 + 4
•
12376 123 + 7 + 6 = . 85904 8 × 5 + 904
•
13064 1 + 3 × 06 + 4 = . 78952 78 + 9 + 52
•
13968 (1 + 39) × (6 + 8) = . 74205 7 × (420 + 5)
•
1079 10 + 7 + 9 = . 456832 (4 + 5 × 68) × 32
•
12397 (1 + (2 + 3) × 9) × 7 = . 46508 4 × 6 × 50 + 8
•
13065 (1 + 3) × 065 = . 29748 2 × (9 + 7 × 4) × 8
•
13986 1×3×9×8+6 = . 74025 (7 + 40) × 25
13065 13 × (0 × 6 + 5) • = . 97284 (9 + 7 × 2 × 8) × 4
•
14063 1 × 40 + 6 + 3 = . 78925 (7 + 8) × 9 × 2 + 5
•
14067 1 + 40 × (6 + 7) = . 29538 2 × (9 + 538)
•
1 + 40 × (6 + 7) 14067 = . 39528 3 × (9 + 52) × 8
•
14067 1 + 40 × (6 + 7) = . 39852 3 × (98 × 5 + 2)
10854 10 × (8 + 5) + 4 • = . 62937 (6 × 2 × 9 + 3) × 7
•
1 + 08 + 5 + 4 10854 = . 72963 7 + 2 × (9 × 6 + 3)
12397 1+2+3+9+7 • = . 58604 5 × 8 + 60 + 4
•
10857 1 × (08 + 5) × 7 = . 49632 4 × 96 + 32
•
•
10857 1 × 08 × 5 + 7 = . 63294 6 × (3 + 2) × 9 + 4
•
•
1 + 09 × 2 × 3 10923 = . 75468 7 + 5 + 46 × 8
•
•
•
•
•
13072 1 + 3 + 072 = . 59684 5 + 9 × (6 + 8 × 4) 13247 1 × 3 × (2 + 4 + 7) = . 96805 9 + 6 × 8 × 05
•
13294 (1 + 3 × (2 + 9)) × 4 = . 57086 570 + 8 + 6
•
14076 1 × 4 + 07 × 6 = . 52938 5 × (2 + 9) × 3 + 8
•
12684 1×2+6+8×4 = . 53907 5 × (3 × 9 + 07)
•
13428 1 × 3 × (4 + 2 × 8) = . 57069 5 × (7 × 06 + 9)
•
14076 14 + 0 × 7 + 6 = . 59823 5 × 9 + 8 × (2 + 3)
10934 109 + 34 = . 78526 7 + 85 × 2 × 6
•
1×2+7×3×6 12736 = . 58904 (58 + 90) × 4
•
1×3×4×8+2 13482 = . 59706 5 + 9 + 70 × 6
•
14256 1 + 425 + 6 = . 70983 (709 + 8) × 3
10935 1 × 09 × 3 × 5 = . 24867 (2 + 48) × 6 + 7
•
12789 12 + (7 + 8) × 9 = . 43065 430 + 65
•
13572 1 + 3 + 5 × 72 = . 49068 (4 + 90) × (6 + 8)
•
14325 1 × 4 + 3 × (2 + 5) = . 87096 8 × 7 + 096
•
1 × 09 × 3 × 5 10935 = . 27864 (2 + 78 + 6) × 4
12789 1+2×7+8×9 • = . 46305 4 + 6 + 305
14508 1 × 4 + 50 + 8 = . 36972 (3 + 69 + 7) × 2
•
1 × 4 + 50 + 8 14508 = . 67392 6 × (7 + 39 + 2)
•
14592 (1 + 4 + 59) × 2 = . 76038 7 + 60 × (3 + 8)
13608 (1 + 3) × (6 + 08) = . 49572 (4 + (9 + 5) × 7) × 2
•
14679 1 + 4 × (6 + 7) × 9 = . 50328 50 × 32 + 8
•
•
1 + 09 + 35 10935 = . 47628 4 × 7 × 6 + 28
•
•
10962 (10 + 9) × 6 + 2 = . 45738 4 + (57 + 3) × 8
•
•
12798 1 + 2 + 79 + 8 = . 60435 60 × (4 + 3) + 5
13572 1×3×5+7×2 • = . 94068 9 + 4 × 06 × 8
•
13608 (1 × 3 + 60) × 8 = . 27945 (2 × 7 + 9) × 45
1 × 2 × (8 + 5 × 7) 12857 = . 49036 4 + 9 × 036
•
•
12903 1 + 2 × 90 + 3 = . 47685 (4 + 7 + 6) × 8 × 5
•
13608 1 + 3 × 60 + 8 = . 94752 94 × (7 + 5 + 2)
•
14703 1 + 4 × (7 + 0 × 3) = . 68952 (6 + 8) × 9 + 5 × 2
10962 (10 + 9) × 6 + 2 = . 57834 578 + 34
•
12903 1 + (2 + 9) × 03 = . 48576 4 × (8 + 5) + 76
•
13608 (1 + 3) × (6 + 08) = . 95742 (9 + 5) × 7 × 4 + 2
•
(1 + 4) × (70 + 6) 14706 = . 83592 8×3×5×9×2
10965 10 + (9 + 6) × 5 = . 23478 2 + 3 × 4 × (7 + 8)
12908 12 + 9 + 0 × 8 = . • 53476 53 + 4 × 7 + 6
13608 (1 + 3) × (6 + 0 × 8) • = . 97524 (9 + 75) × 2 + 4
•
14756 1 + 4 + 7 + 56 = . 20398 2 × (039 + 8)
•
1476 (1 + 4) × 7 + 6 = . 390528 3 × (90 × 5 + 2) × 8
•
14952 1 × 4 × (9 + 5 + 2) = . 80367 8 × (036 + 7)
•
1×4+9+7+6 14976 = . 30528 3 + 05 × (2 + 8)
•
14985 (1 + 49) × 8 + 5 = . 27306 2 + 730 + 6
•
14985 1 + 49 + 85 = . 60273 60 × (2 + 7) + 3
10965 10 + 9 × 65 • = . 27348 2 × (734 + 8)
•
12956 1 + 2 × (9 + 5 × 6) = . 38704 3 + 8 × 7 × 04
•
13708 1 + 3 × (7 + 08) = . 96254 9 + 62 × 5 + 4
12963 1 + 2 × (9 + 63) = . 54087 5 + 40 × (8 + 7)
•
•
12964 1×2+9+6×4 = . 53708 5 × (3 × 7 + 08)
•
13872 1 + 3 + (8 + 7) × 2 = . 95064 9 + (50 + 6) × 4
•
1 × 30 × 6 + 4 13064 = . 59782 (59 × 7 + 8) × 2
•
(1 + 3 + 9 + 6) × 5 13965 = . 87024 (8 + 70 × 2) × 4
10965 10 + (9 + 6) × 5 = . 78432 (7 + 8 + 4) × 32
•
•
10975 (1 + 09) × 7 + 5 = . 32486 3 × (2 + (4 + 8) × 6)
•
(1 + 2) × 096 12096 = . 35784 3 × (5 × 7 × 8 + 4)
•
1 × 2 + 48 × 3 12483 = . 70965 (70 + 96) × 5
•
12584 (1 + 2) × 5 + 84 = . 90376 9 × (03 + 76)
10935 1 + 09 + 35 = . 42768 4 × 27 + 68
•
12483 1×2×4+8+3 = . 70956 7 + 095 + 6
13824 13 × (8 + 2 × 4) = . 50976 (50 + 9) × (7 + 6)
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
15048 1 + 50 × 4 + 8 = . 29376 2 × (9 × 3 + 7) × 6
•
15408 15 × (40 + 8) = . 97263 9 + 72 × 63
•
16359 1 + 63 + 59 = . 20748 2 × 074 + 8
•
17582 17 + 5 × 8 + 2 = . 30694 3 + 06 + 94
•
15048 (1 + 50 + 4) × 8 = . 37962 37 × (9 + 6) × 2
•
15687 1 + (5 + 6) × (8 + 7) = . 30429 304 + 2 × 9
•
16524 1 + 6 + 52 + 4 = . 97308 9 × 7 + 308
•
17589 1 × 75 + 89 = . 32604 (3 + 2) × 60 + 4
1 + 50 + 48 15048 = . 39672 3 × (9 + 6 + 72)
•
1×5×6+9+8 15698 = . 20374 20 + 37 + 4
•
1 × 6 + (5 + 8) × 7 16587 = . 93024 9 × 30 × 2 + 4
•
•
(1 + (7 + 6) × 8) × 2 17682 = . 54309 5 × (4 × 30 + 9)
•
17802 17 × 8 + 02 = . 49536 (49 + 5 × 3) × 6
•
(1 × 8 + 07) × 5 18075 = . 29643 2 × (9 + 6) × 4 + 3
•
18096 1 × 8 + 096 = . 52374 5 + 2 × 37 × 4
•
18239 1+8+2+3+9 = . 67405 (6 + 7 + 4) × 05
•
18 + 27 + 5 18275 = . 40936 40 + (9 + 3) × 6
•
18297 1 × 8 + 2 + 97 = . 40356 4 × (03 + 56)
•
18326 (1 + 8 × 3) × 2 + 6 = . 40579 40 + 5 + 79
15048 (15 + 04) × 8 • = . 76923 7 × (6 × 9 × 2 + 3)
15708 1 + 5 × 7 + 08 • = . 49623 4 + 9 × (6 × 2 + 3)
16704 16 × 7 × 04 • = . 25839 (2 + 5) × (8 + 3) × 9
15067 1 + 5 × 06 + 7 = . 34892 (3 + 4 × 8 + 9) × 2
15732 1 × (5 + 7) × 3 + 2 • = . 40986 4 + 09 + 86
16758 1 + 6 + 7 × (5 + 8) • = . 93024 9 × 30 × 2 + 4
15732 1 × (5 × 7 + 3) × 2 • = . 96048 9 × (6 + 04) × 8
16807 (1 × 6 + 8) × 07 • = . 94325 (9 × 4 × 3 + 2) × 5
•
•
15067 1+5+0×6+7 = . 89243 (8 + 9) × 2 + 43 1 + 50 + 9 × 3 15093 = . 24768 24 + (7 + 6) × 8
•
•
15093 1 + 50 + 9 × 3 = . 27864 2 + 78 + 64
•
•
1 + (5 + 09) × 3 15093 = . 78624 7 × ((8 + 6) × 2 + 4)
•
•
15736 1 × 5 + 7 + 36 = . 90482 (90 + 48) × 2
•
16872 16 + 8 × 72 = . 39045 (3 × 90 + 4) × 5
(1 + 58 + 7) × 3 15873 = . 62049 (6 + 20 × 4) × 9
•
•
15876 1 + 58 + 7 + 6 = . 30429 30 × 4 + 2 × 9
•
17028 170 + 28 = . 49536 4 × (9 + 5 × 3) × 6
15246 (1 + 5) × 2 + 4 + 6 = . 90783 (9 + 07) × 8 + 3
•
15876 1 × 5 + 87 + 6 = . 30942 3 + 094 × 2
•
17034 17 × (0 × 3 + 4) = . 92685 ((9 + 2) × 6 + 8) × 5
•
1 × 8 + 6 × 03 18603 = . 52947 (5 + 2) × 9 + 4 + 7
•
1 × 5 + 30 + 8 15308 = . 64792 6 × 4 + 79 × 2
•
(1 + 5) × 9 + 03 15903 = . 46872 4 × (6 + 8 + 7) × 2
•
17052 17 × 05 + 2 = . 69384 6 × (9 × 3 + 8 × 4)
•
18643 1 × 8 × 64 + 3 = . 70952 70 × (9 + 5) × 2
•
15309 15 + 3 × 09 = . 24786 2 + 4 × (7 + 8) + 6
•
15974 15 + 9 + 74 = . 28036 2 × (80 + 3) + 6
•
17 × 05 + 2 17052 = . 83496 (8 + (3 + 4) × 9) × 6
•
18675 1+8+6+7×5 = . 49302 (4 × 9 + 30) × 2
•
15984 1 + 59 + 84 = . 60273 60 × (2 + 7) + 3
17068 (1 + 7) × 068 • = . 43925 (4 + 3 × 92) × 5
•
1 × 8 × 7 + 03 18703 = . 26945 2 + 6 × (9 + 4) + 5
•
18703 1 × 8 × 7 + 03 = . 92564 9 × (2 + 5 × 6) + 4
•
18705 1 × 87 × 05 = . 24639 2 + 4 + 63 × 9
•
18706 1 + 87 + 06 = . 49352 4 × ((9 + 3) × 5 + 2)
•
18942 1 × (8 + 9 + 4) × 2 = . 70356 (7 × 03 + 5) × 6
15309 (1 + 5) × 30 + 9 • = . 24867 (2 + 48) × 6 + 7
•
•
(1 + 5) × 30 + 9 15309 = . 27864 (2 + 78 + 6) × 4 15309 1 + 53 + 09 = . 42768 4 × 27 + 68
16027 1 + 6 × (02 + 7) = . • 49538 (49 + 5) × 3 + 8
•
15309 (1 + 5 + 30) × 9 = . 62748 (6 × 27 + 4) × 8
•
15309 (1 + 5 + 3) × 09 = . 67284 (6 × 7 + 2) × 8 + 4
•
•
15309 (1 + 53) × 09 = . 78624 78 × (6 + 2) × 4
•
1 × 5 × (3 × 2 + 7) 15327 = . 89604 (89 + 6) × 04
•
•
16038 1 × 6 × (03 + 8) = . 27459 2 × (7 + 45) + 9 16038 160 + 38 = . 27945 (2 + 7 × 9 + 4) × 5
•
•
•
1×6×9+7×4 16974 = . 32085 (3 + 20 + 8) × 5
17 × (0 × 8 + 5) 17085 = . 26934 2 + (6 + 9 × 3) × 4 17085 17 × (0 × 8 + 5) = . 29346 2 × (9 × 3 + 46) 1 + 7 + 09 × 4 17094 = . 52836 5 × (2 + 8 × 3) + 6
16038 160 + 38 = . 74925 (7 × 4 + 9) × 25
•
17253 1 + 72 + 5 + 3 = . 86904 8 × 6 + 90 × 4
•
18952 (1 + 8 + 9) × 5 + 2 = . 40376 40 × 3 + 76
•
16038 1 × 6 × (03 + 8) = . 95742 (9 + 5) × 7 × 4 + 2
•
17256 (1 + 7) × 2 + 56 = . 40983 (40 + 9 + 8) × 3
•
18954 1 + 8 × (9 + 5) + 4 = . 20736 2 + 07 × 3 × 6
•
160 + 7 × 4 16074 = . 82935 (8 + 2 × 93) × 5
•
17298 (1 + 7 + 2) × 9 + 8 = . 60543 60 × 5 + 43
•
1×8+9+7×2 18972 = . 56304 (5 + 6 × 3) × 04
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
19035 (1 + 90 + 3) × 5 = . 24786 (2 × 47 + 8) × 6
•
19824 19 + (8 + 2) × 4 = . 30576 3 × 05 + 76
•
19035 (1 + 90 + 3) × 5 • = . 42768 4 × (27 + 6) × 8
•
19836 (1 + 9 × 8 + 3) × 6 = . 27405 2 × 7 × (40 + 5)
20536 20 + (5 + 3) × 6 • = . 41978 4 + 1 × 9 × (7 + 8)
19038 (1 + 9) × 03 + 8 • = . 72645 7 + 2 × (64 + 5)
19836 1 × 98 + 3 × 6 • = . 47025 (4 + 7) × 025
•
•
19072 (1 + 9 × 07) × 2 = . 34568 (3 + 4 × 5 + 6) × 8
•
19845 (1 + 9) × 8 × (4 + 5) = . 67032 (6 + 70) × 32
20493 2 + 04 × 9 × 3 = . 87561 8 + 7 × (5 + 61)
•
21793 2 × (1 × 7 + 9 + 3) = . 50468 5 × 04 + 68
•
21978 2 × (1 + 97 × 8) = . 34056 (3 + 40) × 56
•
23085 2 + 3 + 08 × 5 = . 47196 4 × (7 + 1 + 9 + 6)
•
23085 2 + 3 + 08 × 5 = . 49761 (4 + 9) × 7 + 6 × 1
•
23175 2 × (3 + 1 × 7) × 5 = . 96408 96 + 40 × 8
20654 2 × (065 + 4) = . 31879 3 × 1 × (8 + 7 × 9)
•
2 × 3 + 49 + 5 23495 = . 61087 6 + 10 × (8 + 7)
20541 (2 + 05) × 41 • = . 93687 93 × (6 + 8) + 7
•
2 + 06 + 18 20618 = . 93574 9+3×5×7+4
19872 1 + 98 + 7 + 2 = . 43056 (4 + 30 + 5) × 6
•
•
(1 + 9 + 8 × 7) × 2 19872 = . 56304 (5 + 6) × (30 + 4)
•
20691 2 + 06 × 9 + 1 = . 45738 45 + 73 + 8
•
23504 2 + 3 × 50 + 4 = . 78196 (7 × 8 + 1) × 9 + 6
19278 1 × 9 × (27 + 8) = . 56304 5 × (6 × 30 + 4)
•
20163 20 × (1 + 6) + 3 = . 59784 ((5 + 9) × 7 + 8) × 4
•
20736 2 + 07 × 3 × 6 = . 91854 9 × (1 + 8 + 54)
•
23598 (2 + 3 + 5 + 9) × 8 = . 41607 4 × 1 × (60 + 7)
19305 (1 × 9 + 30) × 5 • = . 46728 4 × (6 + 7 × 2 × 8)
•
20169 20 + (1 + 6) × 9 = . 47385 (4 × 7 + 3 + 8) × 5
•
20851 2 + 085 × 1 = . 69743 69 + 74 × 3
•
23716 (2 + 3) × 7 × (1 + 6) = . 59048 5 × (90 + 4 × 8)
19305 (1 × 9 + 30) × 5 • = . 76824 768 + 2 × 4
2 × (01 + 7 × 8) 20178 = . • 93456 (9 + 3) × 4 × (5 + 6)
20896 (2 + 08) × 9 + 6 = . 43751 4 × (3 + 7) × 5 + 1
•
•
23718 2 × (3 + 7 × 1 × 8) = . 54069 5 × 40 + 69
•
23856 (2 + 3 + 8) × 5 + 6 = . 97104 9 + 7 × 10 × 4
•
24058 2 + 4 + 05 × 8 = . 93617 9 × 3 × 6 + 17
20987 20 + 9 × (8 + 7) = . 64315 (64 + 31) × 5
•
24381 2 + 4 + 381 = . 60795 60 × (7 + 9) + 5
19203 19 + 203 = . 64875 (6 + 4) × (8 + 7) × 5
•
•
1×9×2+5+6 19256 = . 70384 70 + 3 × (8 + 4)
•
•
19305 1 × (9 + 30) × 5 = . 76824 768 + 2 × 4
•
•
19305 (19 + 3) × 05 = . 78624 7 × (8 + 6 + 2) × 4
•
•
•
2 × (09 + 5 × 8) 20958 = . 64371 6 × (43 + 7) + 1
20196 (2 + 01 × 9) × 6 = . 34578 (3 + 4) × 5 + 78
•
•
20196 2 × (01 + 9 × 6) = . 57834 5 × (7 + 8 × (3 + 4))
•
(1 × 9 + 30) × 5 19305 = . 86427 864 + 2 + 7
•
20356 2 × (0 × 3 + 56) = . 97418 (9 × 7 + 4 × 1) × 8
•
21045 210 + 4 × 5 = . 98637 (9 + 8) × 63 + 7
•
24 × (65 + 1) 24651 = . 38097 3 × (809 + 7)
19467 1 × 9 × 4 + 67 = . 23058 2 × (3 + 058)
•
2 × 03 × (5 + 8) 20358 = . 46719 46 + 7 × 19
•
21056 2 × (1 + 05) × 6 = . 97384 9 + (73 + 8) × 4
•
24651 2 + 4 + 6 + 51 = . 90387 (9 + 03 × 8) × 7
•
24786 2 × (4 + 7) + 8 + 6 = . 95013 (9 × 5 + 01) × 3
•
(24 + 7 × 9) × 5 24795 = . 31806 3 × (180 + 6)
•
24897 2 × (4 + 89) + 7 = . 31605 (3 + 1) × 60 + 5
•
25197 2 × (5 + 1 + 9) + 7 = . 46308 4 × (6 + 3 + 08)
•
25389 (2 + 5 + 3 × 8) × 9 = . 64701 6 + 4 + 701
•
25467 (2 × 5 + 4) × 6 + 7 = . 80319 8 + 031 × 9
•
2 + 5 + 6 × (4 + 1) 25641 = . 90783 (9 + 07) × 8 + 3
1 × 9 × 4 + 67 19467 = . • 80325 (80 + 3 + 2) × 5
20367 2 × 03 + 67 • = . 45198 (4 + 5) × (1 + 9 + 8)
21076 2 + 1 × 07 × 6 • = . 43589 4+3×5+8×9
19642 1 × 964 + 2 • = . 35807 3 × (580 + 7)
2 × 03 + 67 20367 = . • 95418 9 + 5 + 41 × 8
21384 2 + 1 × 38 × 4 = . • 70956 7 + 09 × 56
•
19706 1 + 9 × (7 + 06) = . 25384 (2 × 5 × 3 + 8) × 4
•
19728 19 × 7 × 2 + 8 = . 34056 (3 + 40) × (5 + 6)
•
•
(19 + 7) × 3 × 4 19734 = . 25806 2 + 5 × 80 + 6
•
19764 (19 + 7 × 6) × 4 = . 32805 3 + 2 + 80 × 5
•
20376 2 + 03 × 7 × 6 = . 89145 8 × (9 + 1 + 4) × 5
•
21385 (2 + 1 × 3 + 8) × 5 = . 40796 4 × 07 + 96 21408 2 × (1 × 40 + 8) = . 76935 (7 × 6 + 9 × 3) × 5
20398 2 + (03 + 9) × 8 = . 45167 4 × 51 + 6 + 7
•
•
20398 2 × 03 × 9 + 8 = . 65471 6 × (5 + 4 × 7) + 1
•
21509 21 + 50 × 9 = . 78364 78 × (3 × 6 + 4)
•
20493 (20 + 49) × 3 = . 57816 578 + 1 × 6
•
(2 + 1 + 6) × 9 + 3 21693 = . 80574 8 × (05 × 7 + 4)
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
25697 2 × (5 + 6 + 9) × 7 = . 40381 40 × (3 + 8 × 1)
•
27405 (2 + 7 + 40) × 5 = . 86913 86 × 9 × 1 + 3
•
29058 2 × 9 + 05 × 8 = . 71643 7 × (16 + 4) + 3
•
25704 2 + 57 + 04 = . 39168 3 × 9 + 1 + 68
•
27458 2×7+4×5+8 = . 96103 9 × (6 + 10) + 3
•
29078 2 + 9 + 07 × 8 = . 45136 4 × (5 × (1 + 3) + 6)
25 + 7 + 04 25704 = . • 81396 (8 + 1 + 3) × 9 + 6
27495 (2 + 7 × 4 + 9) × 5 • = . 36801 3 × (6 + 80 + 1)
25704 (2 × 5 + 7) × 04 = . 83916 8×3×9×1+6
•
•
25803 2 + 5 × 8 × 03 = . 64719 (6 + 4 × 7 × 1) × 9
•
•
•
(2 + 9) × (1 + 06) 29106 = . 57834 57 + 8 × 3 × 4
30195 30 × (1 + 9) + 5 = . 46728 4 × (6 + 7 × 2 × 8)
•
30195 30 × (1 + 9) + 5 = . 76824 768 + 2 × 4
•
30195 30 × (1 + 9) + 5 = . 86427 864 + 2 + 7
•
30195 30 × 1 × 9 + 5 = . 86742 8 × 6 + 742
•
30264 3 × 026 × 4 = . 51798 517 + 9 + 8
(2 + 7 × 4 + 9) × 5 27495 = . 68103 6 × 8 × 10 + 3
•
•
27531 2 × (7 + 5 × 3 + 1) = . 40698 4 × (06 + 9) + 8
•
29106 2 + 9 × 10 + 6 = . 83457 8 + (34 + 5) × 7
•
30267 30 × (2 + 6) + 7 = . 58941 (5 + 8) × (9 × 4 + 1)
25806 2 × (5 × 8 + 06) = . 43197 (4 × 3 + 1 + 9) × 7
•
27594 2+7+5×9×4 = . 38106 3 × (81 + 06)
•
29187 (2 + 918) × 7 = . 30456 30 × 4 × 56
•
30 + 2 × 9 × 4 30294 = . 81675 (8 × 1 × 6 + 7) × 5
25839 2 × 5 × (8 × 3 + 9) = . 40716 40 × (7 + 1 × 6)
•
27639 2 + 7 × (6 + 3) + 9 = . 41085 (4 + 10 + 8) × 5
•
29187 (29 + 18) × 7 = . 40365 (4 + 03) × 65
•
30429 304 + 2 × 9 = . 86751 867 + 51
2 × 9 + 3 + 46 29346 = . • 70518 7 × (05 + 18)
•
30492 (3 + 04) × (9 + 2) = . 51876 5 × (18 + 7) + 6
•
30492 (30 + 4) × 9 + 2 = . 57816 578 + 1 × 6
•
30492 30 × (4 + 9) + 2 = . 81675 (8 + 1 × 6) × 75
•
30495 3 × (0 × 4 + 95) = . 82176 8 × 2 × (1 + 7) × 6
25839 25 + 8 × (3 + 9) • = . 61074 6 + 10 × 7 × 4
•
29106 2 × 91 + 0 × 6 • = . 45738 4 × 5 + 7 × 38
•
2 + (6 + 1) × 4 + 5 26145 = . 90387 90 + 3 × 8 + 7
2 × (7 × 8 + 1 + 9) 27819 = . • 35406 3 × 54 + 06
•
26394 2 × (6 + 3) × 9 + 4 = . 57081 5 × 70 + 8 + 1
•
•
26785 2 × (6 × 7 + 8 + 5) = . 30194 30 + 1 × 94
•
27846 2×7+8×4+6 = . 31059 3 + 10 + 5 × 9
•
29358 (2 + 9) × 3 + 58 = . 67104 (6 × 7 + 10) × 4 29376 2 × (9 + 3) × (7 + 6) = . 50184 501 + 8 × 4
27936 2 × (7 + 9) × (3 + 6) = . 40158 401 + 5 + 8
•
•
2 × (7 + 9) × (3 + 6) 27936 = . 48015 480 + 15
•
29403 294 + 03 = . 57816 578 + 1 × 6
•
3 + 06 + 94 30694 = . 57812 ((5 + 7) × 8 + 1) × 2
2 × (68 × 7 + 1) 26871 = . 43095 (4 + 30) × 9 × 5
•
28063 28 × 06 + 3 = . 91574 9 × (1 + 57 + 4)
•
2 × (9 × 6 + 01) 29601 = . 75348 7 × (5 + 3 + 4 × 8)
•
30721 3 + 07 + 21 = . 58469 5×8+4+6+9
•
27018 2 × (70 + 1 + 8) = . 65493 6 + 5 + 4 × 93
•
28341 2 + (8 + 3) × 4 + 1 = . 56079 5 × 6 + 07 × 9
•
29631 (2 + 9 + 6) × (3 + 1) = . 85407 (8 + 5 × 4) × 07
•
30814 3 × (08 + 1) + 4 = . 67592 6 + 7 + 5 × (9 + 2)
•
27054 2 + 7 + 05 + 4 = . 91683 9+1+6×8+3
28457 2 + 84 × 5 + 7 = . 36019 3 + 60 × 1 × 9
•
2 + 9 + (6 + 3) × 7 29637 = . 84105 (8 × 4 + 10) × 5
•
•
30825 3 × (08 + 2) × 5 = . 96174 9 × (6 × (1 + 7) + 4)
•
30846 308 + 4 + 6 = . 71295 7 × (12 + 9) × 5
•
30927 3 × 09 × 2 + 7 = . 48165 (4 + 8 + 1 + 6) × 5
•
30942 3 + 094 × 2 = . 75168 (75 + 1) × 6 + 8
•
31027 31 + 0 × 2 + 7 = . 86549 (8 + 6) × 5 + 4 × 9
•
31065 (3 + 10 + 6) × 5 = . 98427 (9 + 8 × 4 + 2) × 7
•
3 × 12 + 06 31206 = . 95847 9 × 5 + (8 + 4) × 7
•
2 + 7 × 08 + 1 27081 • = . 46359 4 × (6 × 3 + 5) + 9
•
27081 2 + 70 + 81 = . 93456 (9 + 3) × 4 × (5 + 6)
28576 (2 + 8 + 5) × 76 = . • 39104 39 × 10 × 4
•
•
29673 2 × (9 + 6 × 7) + 3 = . 80541 80 + 5 × 41 29678 2+9×6+7+8 = . 30514 3 + 05 × 14 29835 (2 + 98) × (3 + 5) = . 71604 (7 + 1) × 60 × 4
28917 2 × (8 × 9 + 1) + 7 = . 46305 (46 + 3) × 05
•
•
28945 (2 × 8 + 9) × 4 + 5 = . 60371 6 + 03 × 71
•
30128 (30 + 12) × 8 = . 95764 (9 + 5) × 76 + 4
•
2+8+9×7+1 28971 = . 43065 (4 + 3 × 06) × 5
•
3 × 01 × 86 30186 = . 49725 (4 + 9 + 72) × 5
2709 2 × (7 + 0 × 9) = . 186534 186 × 5 + 34
•
•
27189 2 × 71 + 8 + 9 = . 40356 4 × (03 + 56)
•
2 + 7 × (3 + 9 × 1) 27391 = . 58604 (5 × 8 + 6) × 04
•
28695 2×8+6×9+5 = . 40173 (4 + 01) × 7 × 3
•
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
31209 3 + 1209 = . 47586 (4 × 75 + 8) × 6
•
32508 3 × (2 + 5) × 08 = . 74691 7 × (46 + 9) + 1
•
36108 3 + (6 + 1) × 08 = . 49572 4 × 9 + 5 × (7 + 2)
•
31248 3 × (1 + 2 + 4) × 8 = . 59706 5 × 9 × 7 + 06
•
32509 32 × (50 + 9) = . 41876 4 × 1 × 8 × 76
•
36108 3 + 6 × 1 × 08 = . 59472 (5 + 9 + 4 × 7) × 2
•
•
31 × (2 × 4 + 8) 31248 = . 60795 60 × (7 + 9) + 5 31407 3 × 1 × 4 + 07 = . 92568 9×2+5×6+8
•
•
•
(3 + 8) × 016 38016 = . 45792 4 × (5 × 7 + 9 × 2)
•
38165 (3 + 8 × 1 × 6) × 5 = . 70942 (70 + 9) × (4 + 2)
•
38214 (3 + 8) × 2 × 1 × 4 = . 60795 60 + (7 + 9) × 5
•
3 × 8 × 6 + 17 38617 = . 52049 52 × 04 + 9
32901 329 + 01 = . 47856 4 × (7 + 8 + 5) × 6
•
36208 3 × 62 × 08 = . 41975 (4 + 19) × 75
•
38625 (38 + 62) × 5 = . 41097 4 × (10 + 9) × 7
31806 31 + 8 × 0 × 6 = . 75924 7 + 59 + 2 × 4
•
(3 + 40) × (5 + 6) 34056 = . 71928 71 + 928
•
36504 3 × 65 × 04 = . 91728 (9 + 1) × 7 × 28
•
38907 3 + 8 × (9 + 07) = . 64152 64 + 152
31842 3 + (1 + 8 + 4) × 2 = . 79056 7 + 9 + 056
•
34056 (3 + 40) × (5 + 6) = . 89712 89 × 7 × 1 × 2
•
38916 (3 + 89) × 16 = . 74025 7 × 40 × 2 × 5
•
(3 × (8 + 9)) + 7 + 6 38976 = . 54201 5 + 4 × (20 + 1)
•
39024 3 × 902 + 4 = . 71568 71 × 5 × (6 + 8)
•
3 + 90 × 4 × 2 39042 = . 71658 7 + 165 × 8
31752 (3 + 1) × 7 × (5 + 2) = . 94608 (9 + 4 + 60) × 8
•
•
(3 + 1 × 84) × 2 31842 = . • 96075 (9 + 6) × 07 × 5 31904 (3 + 1 × 9) × 04 = . 57826 5 × (7 + 8) + 2 × 6
34278 3 × 4 × (2 + 7) + 8 • = . 91605 9 + 1 + 60 × 5
•
3 + 45 + 06 34506 = . 97128 (9 + 7 + 1 + 2) × 8
36207 3 + 6 + 20 × 7 = . 98415 9 × (8 × (4 + 1) + 5)
3 + (6 + 7) × (9 + 2) 36792 = . • 51408 51 × (4 + 0 × 8)
•
•
36897 (3 + 6 × (8 + 9)) × 7 = . 51204 5 × 1 × 204 3 × (6 + 9) + 7 + 4 36974 = . 58102 5 + 81 + 02
34821 3 + 48 + 2 × 1 = . 70956 7 + 095 + 6
•
37092 (3 + 7 × 09) × 2 = . 56481 5 × 6 × 4 + 81
•
39057 3 × 90 + 5 + 7 = . 68142 6 + 81 × (4 + 2)
•
35046 35 + 04 × 6 = . 81972 8 × (1 + 9 + 7) + 2
•
37206 37 + 2 + 0 × 6 = . 59148 5 + 9 + 1 × 48
•
39 × 10 + 5 39105 = . 46728 4 × (6 + 7 × 2 × 8)
32016 (3 + 20) × 16 = . 47589 475 + 8 × 9
•
35046 35 + 04 × 6 = . 87912 8 + 7 × (9 + 1) × 2
•
3 × (7 × 2 + 6 + 8) 37268 = . 94501 9 + 4 × (50 + 1)
•
39105 39 × 10 + 5 = . 76824 768 + 2 × 4
32054 3 + (2 + 05) × 4 = . 79618 7×9+6+1×8
•
35216 (35 × 2 + 1) × 6 = . 98704 (9 + 8) × 70 + 4
37468 3 × (7 × 4 + 6 × 8) • = . 91205 (91 + 20) × 5
•
39 × 10 + 5 39105 = . 86427 864 + 2 + 7
•
39168 3 + 9 + (1 + 6) × 8 = . 57024 5 + 70 + 24
•
39204 3 × (9 + 20 + 4) = . 51876 5 × (18 + 7) + 6
•
39204 392 + 04 = . 57816 578 + 1 × 6
31926 (3 + 1) × (9 + 2 + 6) = . 80754 (8 + 07 × 5) × 4
•
•
3 × 1 × 96 + 8 31968 = . 52704 (52 + 70) × 4
•
•
32076 3 × (2 + 07 × 6) = . • 98415 9 × (8 × (4 + 1) + 5)
•
36125 (3 + 6 + 1) × 25 = . 49708 (4 × 9 + 7) × 08
37926 3 × 7 × (9 + 2 × 6) = . 51084 510 + 84
•
•
•
•
36108 3 + (6 + 1) × 08 = . 95472 (9 + 5) × (4 + 7) + 2
•
•
•
•
32805 3 + 2 + 805 = . 61479 6 × (1 + 4 × 7 × 9)
•
3 + 7 + 90 + 5 37905 = . 62814 62 + 8 × 14
3 × (2 + 8 + 05) 32805 = . 79461 7 × 9 + 46 × 1
31652 31 × 6 + 5 + 2 = . 87904 8 × 7 × 9 + 04
•
32805 3 + 2 + 80 × 5 = . 49167 4 + 9 × 1 × 67
•
32085 (3 + 20 + 8) × 5 = . 47196 (4 × 7 + 1 + 9) × 6
358 + 02 35802 = . • 41769 4 × 1 × 7 × (6 + 9)
•
35802 35 + 8 × 02 = . 96174 9 × (6 + 1) + 74
•
•
37518 3 × (7 + 5 + 1) × 8 = . 49062 4 × (90 + 6 × 2) 37518 (3 × 7 + 5) × 18 = . 62049 (6 + 20 × 4) × 9
•
32085 (3 + 20 + 8) × 5 = . 61479 6 × (1 + 47) + 9
•
35802 358 + 02 = . 97461 974 + 6 × 1
•
37521 3 × (7 + 5 × (2 + 1)) = . 64809 6 + (4 + 8) × 09
•
39501 3 + 9 × (5 + 01) = . 82467 8 × (2 × 4 + 6) + 7
•
32175 (3 + 2 × 1) × 7 × 5 = . 60489 60 × 4 + 89
•
36019 3 × 60 + 19 = . 84527 8 + 452 + 7
•
37584 3 × (7 + 5) × (8 + 4) = . 60291 602 + 91
•
39501 3 × (9 + 5 × 01) = . 82764 8 + (2 × 7 + 6) × 4
•
32508 (3 + 2) × 50 + 8 = . 71946 7 + 1 × 94 × 6
•
360 + 8 + 4 36084 = . 71295 7 × (12 + 9) × 5
•
37845 (3 + 7 × (8 + 4)) × 5 = . 60291 602 + 91
•
3 + 9 × (5 + 6) + 7 39567 = . 42108 4 × (21 + 08)
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
39652 (3 + 9 + 6) × 5 + 2 = . 41807 41 + 8 × 07
•
42718 (4 + 2 + 7) × 18 = . 90365 (90 + 3 + 6) × 5
•
48015 480 + 15 = . 63729 6 + 3 + 72 × 9
•
39712 3 × (9 + 7 + 1) × 2 = . 65408 6 × (5 × 4 + 08)
•
42813 4 × (2 × 8 + 1) + 3 = . 56079 5 × 6 + 07 × 9
•
48015 480 + 1 × 5 = . 76923 7 × (6 × 9 × 2 + 3)
40158 401 + 5 + 8 • = . 63729 6 + 3 + 72 × 9 •
40176 (40 + 1 + 7) × 6 = . 85932 8 × (5 × 9 + 32)
42813 4 + (2 + 8) × 13 • = . 67095 6 × 7 × (0 × 9 + 5)
•
40293 40 + 293 = . 67518 6 × (75 + 18)
•
•
40293 40 + 293 = . 81675 (8 × 16 + 7) × 5
•
43056 4 + (3 + 05) × 6 = . 81972 8 + 19 + 72
48192 4 × (8 + 1) + 92 • = . 76053 7 + (60 + 5) × 3
•
•
53217 53 + 21 + 7 = . 84096 8 × 4 + 096
•
53406 5 × (3 + 40) × 6 = . 81972 8 + 1972
•
53482 (53 + 4 + 8) × 2 = . 76109 76 + 109
•
54302 54 + 30 × 2 = . 87169 8 + 7 × (16 + 9)
•
5 × 4 + 30 + 6 54306 = . 89217 8 + (9 + 2 + 1) × 7
•
54736 5 + 4 + 73 + 6 = . 90812 (9 × 08 + 1) × 2
48609 4 + 86 + 09 = . 75123 75 × 1 × 2 + 3
430 + 5 × 6 43056 = . 91728 (91 + 7) × (2 + 8)
•
•
43068 4 + 3 × 06 × 8 = . 79152 (7 + 9) × (15 + 2)
•
49572 4 + (9 + 57) × 2 = . 83106 (8 + 3 × 10) × 6
•
5 × (4 + 9 + 01) 54901 = . 86273 8 + 6 × (2 × 7 + 3)
4 × 03 × 65 40365 = . 91287 9 × 1 × 28 × 7
•
43092 4 × (3 + 092) = . 86751 8 + 6 + 751
•
4962 4 × (9 × 6 + 2) = . 315087 (3 + 1) × 508 × 7
•
54963 (5 × 4 × 9 + 6) × 3 = . 78012 780 + 12
•
40386 40 + (3 + 8) × 6 = . 57912 5 + 7 × (9 + 12)
•
(4 + 3 + 6 + 7) × 5 43675 = . 82109 8 + 2 × 10 × 9
•
4 × (9 + 7 × 3) + 1 49731 = . 52608 (5 × 2 + 6) × 08
•
56238 (56 + 2) × 3 + 8 = . 79104 (7 × 9 + 1) × 04
•
40572 4 × 05 + 72 = . 96138 (9 + 61) × 3 + 8
•
43928 4 × 3 × (9 + 2 + 8) = . 70516 70 × 5 + 16
50184 50 + 1 × 8 × 4 • = . 67932 6 + 7 × (9 + 3 × 2)
•
5 + 63 + 0 × 4 56304 = . 81972 8 + 19 + 72
•
56342 5 × (6 + 3) × 4 + 2 = . 80179 80 + 179
•
57312 573 + 1 + 2 = . 60894 6 × (08 + 94)
•
57461 5+7+4×6+1 = . 82309 8 + (2 + 3) × 09
•
5769 5+7×6+9 = . 103842 1 × 03 × 8 × 42
•
40821 408 + 21 • = . 59376 (5 × 9 + 3) × (7 + 6)
45072 4 × (50 + 7 × 2) • = . 81693 8 × (1 + 6 × 9 + 3)
40 + (8 + 9) × 6 40896 = . 73152 7 × (31 + 5) + 2
•
40976 4 × (0 × 9 + 7 + 6) = . 58312 (5 + 8 × (3 + 1)) × 2
•
•
41025 4 × 10 × 2 × 5 = . 86973 8 × (6 + 97 + 3)
•
•
45108 4 × (5 + 1 + 08) = . 69273 6 + (9 + 2) × 7 + 3
•
•
4 + 90 × (3 + 2) 49032 = . 75168 (7 + 5 × 16) × 8
50193 50 + 1 × 93 = . 78624 7 × ((8 + 6) × 2 + 4) 50 + 2 + 7 + 4 50274 = . 81396 8 × 1 × (3 + 9) + 6
45126 4 + (5 + 1 × 2) × 6 = . 87309 8 × (7 + 3) + 09
•
(5 + 067) × 2 50672 = . 91843 9 + 1 × 84 × 3
•
45927 45 + 9 × (2 + 7) = . 83106 (8 + 3 × 10) × 6
•
51268 5+1+2+6×8 = . 97043 9 × 7 + 043
•
57694 5 × (7 + 6 + 9 + 4) = . 82103 82 + 103
41038 4 × (10 + 3 × 8) = . 67592 (67 + 5 × 9) × 2
•
46197 4×6×1+9×7 = . 82305 (8 + 23) × 05
•
51408 5 × 1 × 4 + 08 = . 67932 6 + 7 + (9 + 3) × 2
•
5 × 8 + 401 58401 = . 93627 (93 + 6 + 2) × 7
41205 41 × 2 × 05 = . • 76983 7 + 69 × (8 + 3)
•
46389 4+6×3+8×9 = . 75012 (75 + 01) × 2
52 × (10 + 4) 52104 = . • 93687 93 × (6 + 8) + 7
•
59048 5 × (9 × 04 + 8) = . 61732 6 + 1 × 7 × 32
•
59182 (5 + 9) × 18 + 2 = . 60347 (60 + 3) × 4 + 7
•
60273 60 × (2 + 7) + 3 = . 84915 8 × (4 + 91) + 5
•
60324 (6 + 03) × 2 + 4 = . 79518 7+9+5+1×8
•
60324 60 + 3 × 24 = . 91857 9 + 185 + 7
•
60358 60 + 3 + 5 × 8 = . 71492 71 + 49 + 2
•
•
41832 4 × 1 × 83 × 2 = . 60795 60 × (7 + 9) + 5
•
46501 4 + 6 + 501 = . 73892 7 × (3 × 8 + 92) 46731 4 + 67 + 3 × 1 = . 82095 (8 + 2 × 09) × 5
•
52143 5 + 2 × 1 × 43 = . 87096 8 × 7 + 096 52836 5+2+8×3+6 = . 97104 (9 + 7 + 1) × 04
(4 + 1 × 9) × 25 41925 = . 80367 (80 + 3 + 6) × 7
•
•
41952 4 × (1 + 9 × 5) × 2 = . 76038 7 + 60 × (3 + 8)
•
47502 (4 + 7 × 5) × 02 = . 96831 9 + 6 × (8 × 3 + 1)
•
53064 5 × (3 + 064) = . 91872 (9 + 1) × (8 × 7 + 2)
•
42108 4 × (21 + 08) = . 76593 76 + 5 × 9 × 3
•
4 × (7 × 6 + 01) 47601 = . 85239 (8 + 5) × 23 + 9
•
5 × (3 + 1) + 6 × 9 53169 = . 80472 8 × (0 × 4 + 7 × 2)
•
•
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
60372 (6 + 037) × 2 = . 89154 8 × 9 + 1 + 54
•
67032 6 × 7 × 03 × 2 = . 98154 (9 × 8 + 1) × 5 + 4
•
72135 7 × 2 × 1 × 35 = . 90684 9 × 068 + 4
•
1034 10 + 3 × 4 = . 265879 2 + 65 × (8 + 79)
•
60528 (6 + 05 + 2) × 8 = . 73914 7 + 3 × (9 + 1) × 4
•
67053 6 + 7 × (05 + 3) = . 82194 (8 + 2 + 1 × 9) × 4
•
73062 7 + 30 + 62 = . 98154 9 + 8 × 15 + 4
•
1034 1 + 0 × 34 = . 296758 29 + 6 × (7 × 5 + 8)
•
60528 (6 + 05 + 2) × 8 = . 91374 9 + 1 × 37 × 4
67095 6 × 7 × (0 × 9 + 5) • = . 82431 (82 + 4) × 3 × 1
•
73146 7 × 3 + 146 = . 80592 8 × (05 + 9 × 2)
•
1035 1×0×3+5 = . 247986 2 × (4 + 7 + 98 × 6)
•
61047 61 + 0 × 4 + 7 = . 82593 8 + 2 × (5 + 9) × 3
67125 (6 × (7 + 1) + 2) × 5 • = . 84309 8 + (4 + 30) × 9
•
74169 (7 × (4 + 1) + 6) × 9 = . 85023 8 × 50 + 23
•
1035 1 × 03 × 5 = . 279864 (2 × 7 × 9 × 8 + 6) × 4
•
7 × 4 + 56 + 2 74562 = . 89301 8 × 9 + 30 + 1
•
1×0×3+5 1035 = . 294768 2 × (9 + 4 + 76) × 8
•
7 + (50 + 3) × 6 75036 = . 98124 (9 + 8) × (1 + 24)
•
1035 10 + 3 × 5 = . 476928 (4 + 76) × 9 × 2 × 8
•
76032 7×6+0×3+2 = . 91584 9+1×5×8+4
•
1035 (10 + 3) × 5 = . 497628 4 + 9 × 7 × 62 × 8
76302 76 + 30 + 2 • = . 81954 8 × 1 × (9 + 5) + 4
•
1035 1×0×3+5 = . 748926 (74 × 8 + 9 + 2) × 6
•
1045 1×0×4+5 = . 267938 2 + (67 + 93) × 8
•
1 + 0 × 47 1047 = . 385296 38 + 5 × (2 + 9) × 6
•
1053 1 + 05 + 3 = . 247689 (24 + 7) × 68 + 9
•
1053 1 + 05 + 3 = . 268749 26 × 8 × (7 + 4) + 9
•
1053 1 + 0 × 53 = . 287469 (2 × 8 + 7 × 4) × 6 + 9
•
1053 (10 + 5) × 3 = . 467298 4 + 67 × 298
•
1053 (1 + 05) × 3 = . 482976 (4 + 82) × (9 + 7) × 6
•
1 + 05 + 3 1053 = . 642798 6 + 4 × 2 × 7 × 98
86352 (8 × 6 + 3) × 5 + 2 = . 97104 9 + 7 × 10 × 4
•
1053 1 + 05 + 3 = . 694278 69 × (4 × 2 + 78)
•
1026 1×0×2+6 = . 738549 7 + (3 + 85) × 49
•
1053 1×0×5+3 = . 746928 7 × (4 × 69 + 28)
61074 6 + 10 × (7 + 4) • = . 95823 (9 + 5) × (8 + 2 + 3)
•
•
•
•
•
61537 (6 + 1 × 53) × 7 = . 98042 (9 + 80 × 4) × 2 62307 6 × 2 × 3 + 07 = . 94185 9 + 4 × (1 + 8 + 5) 62307 6 × (2 + 3 × 07) = . 94815 (9 + 4 × 8 + 1) × 5
•
•
•
•
(6 × 25 + 1) × 4 62514 = . 78039 7 + (80 + 3) × 9
•
63279 (6 + 3 × 2) × 79 = . 84105 84 × (10 + 5)
•
67284 (6 × 7 + 2) × 8 + 4 = . 90153 9 × 01 × 53 68241 6 × 824 + 1 = . 90735 90 × 73 + 5 6 × 824 + 1 68241 = . 95703 95 × (70 + 3) 68 + 3 × (5 + 2) 68352 = . 79104 7 × 9 + 10 × 4 68742 6 × (8 + 7 + 42) = . 91053 9 × 10 × 5 + 3 69471 6×9+4×7+1 = . 80352 80 + (3 + 5) × 2
(76 + 5 × 9) × 3 76593 = . • 81024 8 × (10 + 2) × 4 80 × 7 × 2 + 5 80725 = . • 93641 9 × (36 × 4 + 1)
• •
63495 6 + 34 + 9 × 5 = . 70218 70 + (2 + 1) × 8
•
69741 6 × (9 + 74 + 1) = . 83025 8 × 3 × 025 •
•
63504 6 + 3 × 50 + 4 = . 91287 (9 + 1) × (2 × 8 + 7)
•
6985 6+9+8×5 = . 217043 2 + 1704 + 3 • 70425 7 × (0 × 4 + 25) = . 81693 8 × (16 + 9) + 3
63512 6+3+5+1+2 • = . 70984 7+0×9+8+4
•
64152 6 × (4 + 1 + 5 + 2) • = . 70389 7+0×3+8×9
70623 7 × (06 + 2) + 3 • = . 98154 9×8+1+5+4
•
•
65043 650 + 4 + 3 = . 87219 872 + 1 × 9 65128 (6 + 5 + 1 + 2) × 8 = . 70943 70 + 9 + 43
70914 70 + 9 × 1 × 4 • = . 83625 8 × (3 + 6 × 2) + 5
•
70914 70 + 9 × 1 × 4 = . 85632 8 × (5 + 6 + 3 + 2)
•
•
•
82365 8 + 2 + 36 × 5 = . 97104 (9 + 7) × (10 + 4) 83421 (8 + 3) × 4 + 2 × 1 = . 90675 9 + 06 + 7 × 5 84672 (8 + 4 × 6) × 7 × 2 = . 90153 9 × 01 × 53 85176 8 × (5 + 17 + 6) = . 94302 (94 + 30) × 2 85241 8 × 5 + 2 + 41 = . 90376 9 + 03 + 76
•
65184 (6 + 5 + 1) × (8 + 4) = . 93702 9 × (3 × 7 + 02)
•
71085 7 × (1 + 08) × 5 = . 93426 9 × (34 + 2 × 6)
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1026 1×0×2+6 = . 1054 1 + 0 × 54 934857 ((93 + 4) × 8 + 5) × 7 • = . 729368 72 × 9 + 36 + 8
•
65379 6 + 5 + 3 + 79 = . 80142 (8 × (014)) + 2
•
71094 7 × 10 + 94 = . 82365 8 + 2 + 36 × 5
•
1032 (1 + 03) × 2 = . 578694 (5 + 78) × 6 × 9 + 4
•
(6 + 5 + 8) × 7 × 3 65873 = . 90142 (90 + 1) × (4 + 2)
•
(7 + 1 × 80) × 4 71804 = . 95326 (9 × 5 + 32) × 6
•
1032 (1 + 03) × 2 1059 1+0×5+9 = . • = . 856947 (8 + 5) × (69 + 4) × 7 427836 4 × (2 + 7 × 8 × 3 × 6)
•
1057 1×0×5+7 = . 263948 2 + 6 × (3 + 9 × 4 × 8)
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
1062 10 + 6 + 2 = . 384975 (38 + 49) × 75
•
1098 1 + 0 × 98 1062 1×0×6+2 1308 1 × 3 + 08 = .• = .• = . 2 + 4 + 3 × (7 + 5) × 6 795264 854379 8 × 5 × 4 × (3 + 7) + 9 243756 7 × 952 + 6 × 4
•
1584 1+5+8×4 = . 703296 703 × (2 × 9 + 6)
1395 1 × (3 + 9) × 5 1098 1 + 0 × 98 1×0×6+2 1062 = . •. = . • = • 847602 84 × 7 × (60 + 2) 3 × (4 × (2 + 5) + 76) 873495 (8 × (7 + 3) × 4 + 9) × 5 342576
•
1 × (5 + 8) × 6 1586 = . 720349 (720 + 3) × 49
•
1593 1+5+9+3 = . 628704 6 × 2 × 8 × (70 + 4)
•
1×5+9+3 1593 = . 726408 (7 + 2 × 6) × 408
•
1602 1+6×0×2 = . 539874 5 + (3 × 9 + 8 × 7) × 4
•
1+6×0×2 1602 = . 587934 5 × (8 + 7 × 9) + 3 × 4
1206 1+2×0×6 1407 1+4×0×7 = .• = . 473958 (4 + 7) × 3 + 9 × 5 × 8 682395 (68 + 2 + 3 × 9) × 5
•
1602 1×6+0×2 = . 947583 (9 + 4) × 7 × (5 + 8) × 3
•
1 × 6 × 05 1605 = . 427893 (4 × 2 + 78) × 93
•
1096 1×0×9+6 1308 1+3+0×8 1584 1×5×8+4 = . • = . • = . 758432 (7 + 5) × (8 × 43 + 2) 629475 (6 + 29) × (4 + 7) × 5 209736 20 × 97 × 3 + 6
1397 1+3+9+7 1098 1 + 0 × 98 1062 1 + 0 × 62 = . •. = .• • = 426085 (4 + 2 × 608) × 5 (4 + 7 × 6 × 5 + 3) × 2 954738 (9 × (5 + 4 × 7) × 3) + 8 476532
•
1063 1 + 0 × 63 = . 285947 2 × 85 + 9 × (4 + 7)
•
1098 1 + 0 × 98 = . 534726 5 + 34 × 7 × 2 + 6 1098 1 + 0 × 98 = . 734562 73 × (4 + 5) + 6 × 2
•
1402 1 + 4 × 02 = . 389756 (38 × 9 + 75) × 6 1406 1+4×0×6 = . 327598 3 × (2 + 7) × 5 + 98
1×0×6+4 1064 = . 798532 79 × (8 + 5 × 3 × 2)
•
1065 (1 + 06) × 5 = . 798324 (79 × 83 + 2) × 4
•
•
1069 1 + 069 = . 547328 5 × 4 × 7 × 32 × 8
•
1×2+0×6 1206 14 + 2 + 6 1426 = .• = . 538479 5 + 3 × 8 × (4 × 7 + 9) 980375 9 × 80 × 3 × 7 + 5
•
1072 1 + 0 × 72 = . 693584 6 × 93 + 5 + 84
•
1206 1×2+0×6 = . 973845 (97 × 3 + 8 × 4) × 5
•
1074 1 + 0 × 74 1485 1+4+8×5 1208 1×2+0×8 = .• = . = .• 936528 9 × 3 × (6 × 5 + 2) + 8 754396 7 × 096 × 32 7 + (5 × 4 + 3) × 9 × 6 709632
•
1 + 0 × 83 1083 = . 294576 (29 + 4 + 5) × 7 + 6
•
•
•
•
1245 (1 + 2) × 4 × 5 • = . 890673 (8 + 90) × 6 × 73
1×2+7+3 1273 1083 1×0×8+3 = . = . • 605948 (60 + 59) × 48 947625 (9 × (4 + 7) + 6) × 25 1×0×8+6 1086 = . 539742 (5 + 3 + 9 × 7) × 42
•
1092 1+0×9+2 = . 385476 3 × (85 × 4 + 7 + 6)
•
•
1 + 0 × 92 1092 = . 483756 48 + 3 + 7 × 56
•
•
•
•
•
1+4+5+2 1452 1608 1+6×0×8 = . 370986 (3 × 70 + 9) × (8 + 6) • 794352 = 7 + (94 + 3) × 5 + 2 .
1509 1+5×0×9 1209 1+2+0×9 = . = . • 473826 4 × 73 + 8 × 2 + 6 368745 3 × ((68 + 7) × 4 + 5)
1083 1 + 0 × 83 1209 1+2×0×9 • = .• = . 567492 (5 × 6 + 7 × 4) × 9 + 2 437658 4 + 3 + 7 + 6 × 58 1083 (1 + 08) × 3 • = . 625974 6 × (2597 + 4)
•
1276 1×2+7×6 = . 453908 4 × (5 + 3908)
•
1638 1×6×3+8 = . 240975 (2 + 40 + 9) × 75
•
1638 1+6+3+8 = . 290745 290 × (7 + 4) + 5
•
1638 1+6+3+8 = . 470925 470 × (9 + 2) + 5
•
1523 15 + 2 + 3 = . 409687 4 + 096 × 8 × 7
•
1638 1×6×3+8 • = . 1524 1+5×2+4 972405 9 × 7 × (240 + 5) = . 387096 (3 + 8 × (70 + 9)) × 6 •
1658 1+6+5+8 = . 237094 (2 × 3 + 709) × 4
•
1679 1+6+7+9 = . 308425 (3 + 0842) × 5
•
1683 16 × (8 + 3) = . 240975 2 × 40 × 9 × 7 × 5
•
1692 1×6+9×2 = . 370548 (3 + 70) × (5 + 4) × 8
1547 1+5+4+7 = . 290836 290 × (8 + 3) + 6
•
1704 1+7×0×4 = . 693528 (6 × 9 + 3) × (5 + 2) + 8
•
1726 (1 + 7 + 2) × 6 = . 308954 3 × 0895 × 4
•
1 + 7 + 28 1728 = . 354096 3 × (5 + 409 × 6)
1532 1+5+3×2 = . • 680974 6 + 8 × 09 × 74
•
1539 1+5+3×9 = . 264708 2 × (6 + 4 × 708) 1539 15 + 3 + 9 = . 278046 (2 + 7 + 804) × 6
1293 1×2+9+3 = . 704685 70 × (4 × 6 + 85)
•
•
(1 + 2) × (9 + 6) 1296 = . 350784 3 × (507 × 8 + 4)
•
1092 1 + 09 + 2 = . 657384 (65 + 7 × 3) × 84
•
1305 1 × 3 × 05 = . 972486 (9 + 7 × 2) × 486
•
1547 1+5+4+7 = . 380926 380 × (9 + 2) + 6
1095 1×0×9+5 = . 234768 2 × (3 + 4) × 76 + 8
•
1307 1+3×0×7 = . 845629 8 + (4 + 5 + 62) × 9
•
1+5+7+2 1572 = . 304968 30 × (49 + 6 × 8)
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
1732 (1 + 7 + 3) × 2 = . 465908 4 + 6 + 5908
•
1953 1 + 95 × 3 = . 246078 (2 + 460) × 78
•
2064 2 × 06 + 4 = . 317985 (3 + 17 + 9) × 85
•
1749 17 + 49 = . 386052 3 × 8 × (605 + 2)
•
1956 19 + 5 + 6 = . 320784 3 × 20 × (78 + 4)
•
2109 2 × (10 + 9) 2067 2×0×6+7 = . • 648573 = 6 + 4 × 8 × 5 × 73 . 345189 (3 + 4) × (5 + 18 × 9)
•
1764 1 × 76 + 4 = . 208593 20 × (8 + 5 × 93)
•
(1 + 9) × 7 + 5 1975 = . 462308 462 × (30 + 8)
•
2 + 10 + 9 2109 = . • 2 × 07 + 6 2076 7 × (6 + 3 × 45 × 8) = . 763458 145839 1 + 4 × (5 + 8) × 3 × 9
1782 1+7+8+2 • = . 436095 4360 + 9 × 5 1782 1 + 78 + 2 • = . 540936 (5 + 4093) × 6
•
•
•
1984 1 × 9 + 84 = . 237056 (2 + 370 × 5) × 6
•
2079 2+0×7+9 = . 148365 (148 + 3 + 6) × 5
•
•
2149 (2 + 1) × 4 + 9 = . 683075 (6 + 83 + 0) × 75
•
2153 2×1×5+3 = . 986074 98 × 60 + 74
•
2154 (2 + 1 × 5) × 4 = . 938067 938 × (06 + 7)
•
2178 2 × (1 × 7 + 8) = . 365904 (3 + 6 + 5) × 90 × 4
•
2178 2×1×7+8 = . 436095 4360 + 9 × 5
2014 2 × 01 × 4 2079 2+0×7+9 = •. = . 387695 (3 + 8) × 7 × (6 + 9 + 5) 153468 (1 + 53 + 4) × (6 + 8)
1802 1 + 8 + 02 20 + 1 × 5 2015 = . 679354 (6 + 7) × (9 × 35 + 4) • 879346 = 8 + 79 × 3 × 46 .
•
207 + 9 2079 = . 163548 1 × 6 × 354 × 8
2109 2+1+0×9 = . 364857 3 × (6 + 4 × 8 × 5 + 7)
1+8+0×9 1809 = . 436572 4 × 3 + 6 × 5 × 72
•
1809 1+8+0×9 = . 467325 (4 × 6 + 7) × 3 × 25
•
2079 2×0×7+9 2016 2×0×1+6 = . = . • 846153 (8 + 4) × 61 × 5 + 3 375984 3 + (7 + 5) × (9 + 84)
•
21 + 78 2178 = . 540936 (5 + 4093) × 6
•
1824 (1 × 8 + 2) × 4 = . 305976 305 × (9 + 7 + 6)
•
2084 2 × (0 × 8 + 4) 2016 2×0×1+6 = . = . • 935716 9 + 3 + 5 × 716 753984 (7 + 5 × 3) × (98 + 4)
•
2187 2+1+8+7 = . 304965 30 + 496 × 5
•
(1 + 8) × 2 × 7 1827 = . 930465 930 × (4 + 65)
•
2085 2×0×8+5 2034 2 + 0 × 34 •. = . = 9 × 4 × 61 + 73 519687 (5 + (1 + 9) × 6 + 8) × 7 946173
•
2198 (2 + 1) × 9 + 8 = . 356704 (3 + 5) × (6 + 704)
•
1839 1+8+3+9 = . 765024 7 × 6 × (50 + 2) × 4
•
2034 2 × 03 + 4 = . 716985 (7 + 1 × 698) × 5
•
1854 18 × 5 + 4 = . 267903 2 × 6790 + 3
2095 2×0×9+5 2037 2×0×3+7 •. = . • = 461738 4 + 61 × (7 + 3 + 8) 159468 (1 + 5) × 9 × (4 + 6) + 8
•
1854 1+8+5+4 = . 367092 36 × (7 + 092)
•
•
1872 • = 493506
•
•
•
1873 = 402695 1908 = 423576 1908 = 452673
2079 2+0×7+9 2015 2 × 01 × 5 = . = . • 653184 6 × (5 + 3) × 18 × 4 973648 ((97 + 3) × 6 + 4) × 8
•
2304 2 × (3 + 04) 2×0×8+9 2089 = . • 759168 = 7 × (591 + 68) . 637145 (6 × (3 + 7) + 1) × 45 •
(2 + 30) × 4 2304 = . 791568 (7 + 915 × 6) × 8
2304 • 2097 2 + 0 × 97 795168 • = . 381654 (3 + 8 × 1 × (6 + 5)) × 4 2305 • (1 + 8 + 7) × 2 196847 2 + 1 + 06 2106 . 2041 2×0×4+1 = . • • = . 4 × (9 + 350 × 6) 347958 3 + 4 × 7 × (9 × 5 + 8) 567398 5 + 67 × 3 + 9 × 8 2349 • 175608 1×8+7+3 2106 2 × (1 + 06) 2043 2×0×4+3 . = . • = . • (40 × 2 + 6) × 9 × 5 487539 4 + (8 + 75) × 39 578169 5 × 7 × (8 + 16) + 9 2385 • 410697 1+9×0×8 2106 2 + 10 + 6 2×0×4+5 . 2045 = . = .• 4 + 2 + 3 × (5 + 7) × 6 • 539487 (5 + (3 + 9) × 4) × 87 2394 189367 (1 + 8 × 9 + 3) × 6 + 7 • 107856 1 × 9 × 08 2106 2 + 1 + 06 2046 2 + 0 × 4 + 6 . • = . 2394 • = . (4 + 5) × (26 × 73) 758394 (75 + 8) × 39 + 4 • 835791 83 + 5 × 7 × 91 675108 2037 2 × 03 + 7 • = . 596841 (5 + 9) × 68 × 4 + 1
=
2 × 30 + 4 . (7 + 9 × 51 × 6) × 8
=
2+3+0×5 . (1 × 9 + 6 × 8 + 4) × 7
=
(2 + 3) × 4 + 9 . (1 + 7 × 5) × 60 + 8
=
2 + 3 + 85 . 41 × 06 × 9 × 7
=
2×3+9+4 . 1 × 0 × 7 + 856
=
2+3+9+4 . (6 × 7 + 5) × 108
•
1908 1+9×0×8 = . 675432 6 × (7 + 5 × 4 + 32)
•
2061 2 + 0 × 61 = . 859437 8 + 59 × (4 + 3 + 7)
•
(1 + 9) × (3 + 5) 1935 = . 640872 (6 + 40) × 8 × 72
•
2403 (2 + 40) × 3 2064 2 × (0 × 6 + 4) 2109 2+1+0×9 •. = . = .• = 7 × 1 × 6 × 895 139578 1 + 3 × 9 × (5 + 7 + 8) 345876 (3 + 4 + 5 × (8 + 7)) × 6 716895
•
2106 2 + 1 + 06 = . 795834 79 × (5 × 8 + 3) + 4
•
2394 2×3+9+4 = . 758016 75 × 80 + 16
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
2403 2 + 4 + 03 2703 2+7×0×3 2916 2+9×1×6 = •. = .• = . 768159 7 × (6 + (8 + 1) × 5 × 9) 418965 (4 + 1) × 8 + 9 × 6 × 5 745038 7 × 4 × (503 + 8)
•
2409 24 + 09 = . 136875 (1 × 368 + 7) × 5
2416 2 × (4 + 1) + 6 • = . 798035 7 × (9 × 80 + 35)
•
2 × (4 + 5) + 1 2451 = . 367908 36 × 79 + 08
•
2704 2 + 7 + 04 = . 139568 13 × (9 × 5 + 6) + 8
•
2 × (7 + 0 × 9) 2709 = . 835146 83 × (5 + 1 + 46)
•
•
2476 2 × (4 + 7 × 6) = . 159083 1 × 5908 + 3
•
•
2×7+5×4 2754 2 × 5 + 41 2541 = . = . • 109836 (10 + 9 × 8 × 3) × 6 370986 (3 + 70) × (9 + 8) × 6
•
2+5+4+1 2541 = . 378609 3 × (7 × 8 + 60 × 9)
2568 2 × 5 × (6 + 8) • = . 713904 7 × 1390 × 4
•
2583 (2 + 5) × 8 × 3 = . 410697 4 × 106 × 9 × 7
2934 2 × (9 + 3 × 4) = . 168705 (1 + 68) × 7 × 05
2961 29 + 6 × 1 2705 27 × 05 = . • = .• 38 + 4507 138496 (1 + 3) × 8 × 4 × 9 × 6 384507
2457 245 + 7 = . 103896 (103 + 8) × 96
•
•
•
3051 3 × (05 + 1) = . 687492 6 × (8 + 74 × 9 + 2)
•
3051 3 + 0 × 51 = . 926487 (9 + 26 × 4) × 8 + 7
•
3052 3×0×5+2 = . 614978 6 + 1 + 4 × 97 + 8
•
3068 3 × 06 + 8 = . 157294 1 + (5 × 7 + 2) × 9 × 4
•
3072 3×0×7+2 = . 941568 9 × (4 + 1) + 568
•
3074 30 + 7 × 4 = . 215869 2 + (1 + 58) × 69
2964 (2 + 9 + 6) × 4 = . 137085 1 × 37 × 085 2 × (9 + 6) × 4 2964 = . 301587 30 × (1 + 58 × 7)
2746 2 × (7 + 46) = . 503891 50 × 389 + 1
•
2754 2 + 75 + 4 = . 108936 1 × 089 × 36
•
2967 2 × (9 + 6) + 7 = . 130548 1 × 30 × 54 + 8
•
3 × (0 × 7 + 8) 3078 = . 145692 (1 + 4 × 5) × 6 × 9 + 2
•
(2 + 9 × 7) × 4 2974 = . 130856 130 × 8 × (5 + 6)
•
3078 30 + 78 = . 149625 14 × (9 + 6) × 25
•
3078 3 × (07 + 8) 3015 30 + 1 × 5 = . = . • 156294 1 + (562 + 9) × 4 496872 4 × (96 × (8 + 7) + 2)
(2 × 7 + 5) × 6 2756 = . 394108 39 × (410 + 8)
2796 (2 + 7 + 9) × 6 • = . 108345 (10 + 83) × 45
•
•
•
(2 + 8) × 3 + 5 2835 = . 714906 7 × 14 × 90 + 6
3078 • 3×0×1+5 3015 195624 = . • 742896 (7 + 4) × (2 × 8 + 96) 3078 • 219564 3015 3 × 015 • = . 897264 8 × 9 × (7 × 26 + 4) 3078 • 492651 3024 3 × (02 + 4) • = . 3078 159768 15 + 9 × (7 + 6) × 8 • 615429
=
3 + 07 + 8 . 1 × 95 × 6 × 2 + 4
=
3 + 0 × 78 . 2 × 1 × 95 + 6 × 4
=
3 + 07 + 8 . (4 + 92) × 6 × 5 + 1
=
3 + 07 + 8 . 61 × (5 + (4 + 2) × 9)
2584 (2 + 5 × 8) × 4 = . 610793 61 × 07 × 93
•
2619 2 × (6 + 1 + 9) = . 345708 3 × (4 × 5 × 70 + 8)
•
2843 2×8×4+3 = . 150679 1 × 506 × 7 + 9
•
3024 3 × 024 = . 167958 1 + 6 × 7 × 95 + 8
•
2+6×3+9 2639 = . 158704 1 + 5 × 87 × 04
•
2871 2+8+7+1 = . 390456 (3 + 9 × 045) × 6
•
3024 3 × (02 + 4) 3082 3 + 0 × 82 = . 197568 1 × (9 + 75) × (6 + 8) • 579416 = (57 + 9 × 4 + 1) × 6 .
•
2673 2×6+7+3 = . 105948 (10 + 5 + 94) × 8
•
2871 (2 + 8) × 7 × 1 = . 904365 90 × (43 + 6) × 5
•
3+0×2+4 3024 = . 197856 1 + 9 + 7 × (8 + 56)
•
3084 3×0×8+4 = . 521967 5 + 2 + (1 + 9) × 67
•
2673 2×6+7×3 2871 2 + 8 × (7 + 1) = . 159084 (1 + 5 × (90 + 8)) × 4 • 906453 = 906 × (4 × 5 + 3) .
•
3024 3 × 02 × 4 = . 671958 6 + 7 × (1 + 95 × 8)
•
3087 3 × 08 × 7 = . 269451 26 × 94 × (5 + 1)
•
3087 3×0×8+7 = . 421596 4 × 215 + 96
•
3087 3 + 0 × 87 = . 564921 56 + 492 + 1
•
3087 3×0×8+7 = . 649152 6 × 49 × 1 × 5 + 2
•
•
28 + 39 2839 = . 150467 1 + 50 × (4 + 67)
2673 2×6+7×3 2871 2 + 8 × (7 + 1) • = . 159408 (1 + 5 × (9 + 40)) × 8 • 930465 = 93 × 046 × 5 . 2673 2×6+7×3 = . 491508 4 × (9 + 1508)
•
•
2691 2×6+9×1 = . 387504 3 × 8 + 750 × 4
•
2691 26 + 91 = . 870435 87 × 0435
•
•
3024 3×0×2+4 = . 718956 7 × (18 + 9) × 5 + 6
•
3079 3 + 0 × 79 = . 412586 (41 + 2 × (5 + 8)) × 6
2905 (2 + 90) × 5 = . 148736 1 × 4 × 8 × 736
•
•
2907 2+9×0×7 = . 683145 (6 + 83 + 1 + 4) × 5
•
3042 3×0×4+2 = . 571896 5 × 7 × 1 × 8 + 96
•
3105 3 × 1 × 05 = . 247986 2 × (4 + 7 + 98 × 6)
•
2916 2×9×1+6 = . 708345 70 × 83 + 4 × 5
•
3048 3×0×4+8 = . 167259 1 + 6 × (7 × 2 + 59)
•
3105 3 × (10 + 5) = . 279864 (2 × 7 × 9 × 8 + 6) × 4
3027 3×0×2+7 = . 469185 4 × (6 + 9) × 18 + 5
Inder J. Taneja
•
3105 3 × 1 × 05 = . 294768 2 × (9 + 4 + 76) × 8
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
3204 3 × (2 + 0 × 4) = . 975618 9 × 7 × (5 + 6 + 18)
•
3502 3 + 5 + 02 = . 894761 8 + 9 × (47 × 6 + 1)
•
3672 3 + 67 + 2 = . 184059 18 × 40 × 5 + 9
3701 3 × 7 × 01 • = . 3504 3+5+0×4 (8 × 9 + 5) × (64 + 2) • = . 895642 219876 2 + (1 + 9) × (8 + 7 × 6)
3105 (3 + 1 + 0) × 5 • = . 694278 (69 × 4 × 2 + 7) × 8
3214 3 × (2 + 1 × 4) • = . 965807 9 × (6 + 5 + 80) × 7
3 × 1 × 05 3105 = . • 748926 (74 × 8 + 9 + 2) × 6
3507 3 × 5 × 07 3249 3×2+4+9 = . • = . • 148296 148 × 2 × (9 + 6) 108756 (10 + 8 × (7 + 5)) × 6
•
3705 3 + 7 + 05 = . 149682 (1 + 49 × 6 + 8) × 2
•
3+7+0×8 3708 = . 196524 1 + 9 + 65 × 2 × 4
•
3708 (3 + 7) × 08 = . 264195 2 × 6 × (4 + 1) × 95
3519 3 × (5 + 19) = . 468027 (4 + 680) × 2 × 7
•
3+7×0×8 3708 = . 645192 (6 × 4 + 5 × 1) × 9 × 2
3105 (3 + 1 + 0) × 5 3×2+4+9 3249 = . • = . 942678 (94 × (2 + 6) + 7) × 8 580716 (5 + 80 × 7 + 1) × 6
•
3108 3 + 10 + 8 3 × 2 × (7 + 6) = . 3276 245976 ((2 + 4) × 5 × 9 + 7) × 6• 819504 = 8 + 19504 .
•
•
3 × 1 × 08 3108 3289 3×2+8+9 = •. = . 495726 (4 + 9 × 5 × 7) × (2 × 6) 175604 (1 × 7 + 5 × 60) × 4
•
3528 3 × (5 + 2) × 8 = . 407169 (40 × 7 + 1) × 69
•
3729 37 + 2 × 9 = . 140685 (1 + 406 + 8) × 5
•
3108 3 × (1 + 0 × 8) = . 649572 6 + 49 + 572
•
3567 3 × 5 + 67 = . 129804 1 × 2980 + 4
•
3741 3 × (7 + 4) + 1 = . 692085 (6 × 9 + 20) × 85
•
3108 3+1+0×8 3295 (3 + 2 + 9) × 5 = •. = . 726495 (7 + 2 × (6 + 4) × 9) × 5 168704 16 × 8 × 7 × 04
•
3 × (5 + 7) × 2 3572 = . 108946 (10 + 89 × 4) × 6
•
3768 3+7+6+8 = . 129054 12 + 90 × (5 + 4)
•
3108 3+1+0×8 3+4+0×2 3402 3+5+8+6 = . • = . 3586 964257 (9 + 64) × (2 × 5 + 7) 156978 ((1 + 5) × 6 + 9) × 7 + 8• 104972 = (10 + 4 × 9) × 7 × 2 .
•
3768 (37 + 6) × 8 = . 495021 4 + 9 × 5021
•
3795 3+7+9×5 = . 128064 (1 + 28) × 064
•
(3 × 7 + 9) × 8 3798 = . 102546 10 × 2 × 54 × 6
•
3798 3×7+9+8 = . 140526 140 × 5 × 2 + 6
•
•
•
3 + 129 3129 = . 475608 (4 × 7 + 5) × 608 3129 3 + 129 = . 860475 (8 × 60 + 4) × 75
•
•
3289 3×2+8+9 = . 617045 61 × 70 + 45
3402 3 + 402 = . 197568 (1 + 97) × 5 × 6 × 8
•
3507 3 × 5 × 07 = . 481962 481 × (9 + 6) × 2
3591 3 + 59 + 1 = . 278046 (2 + 7 + 804) × 6
3402 3+4×0×2 = . 718956 7 × 1 × 89 + 5 + 6
•
•
3159 3 × 15 + 9 3408 3 × 4 + 08 = . • = . 240786 (2 + 40) × 7 × (8 + 6) 172956 1 × 7 + 2 × 9 × 56
•
•
3159 3 × (15 + 9) = . 247806 2 + 4 + 7 × 806
•
3409 3+4+0×9 3+6×0×9 = . 3609 167528 (1 × 6 + 7 × 5 + 2) × 8 • 714582 = 7 + 1 + 4 + 582 .
•
3 + 15 + 9 3159 = . 846027 (8 + 4) × 602 + 7
•
3409 3+4+0×9 3618 36 + 1 × 8 = . 851276 (8 + 5 × (1 + 2)) × 76 • 240597 = 2 × (40 × 5 + 9) × 7 .
•
3186 3+1+8+6 = . • 209745 (209 + 7 × 4) × 5
•
3186 3 × (18 + 6) = . 294705 2 × 9 × (4 + 70) × 5
•
3416 3+4+1+6 = . • 905728 (90 × 5 + 7 × 2) × 8
•
3186 3+1+8+6 = . 524097 (5 + 2 + 40) × 9 × 7
•
•
3198 31 + 9 + 8 = . 476502 (4 + 7) × 650 + 2
•
(3 + 20) × 4 3204 = . 716895 (7 + 16) × 895
•
3429 3×4+2×9 = . 568071 5 × (6 + 8 + 0) × 71
3591 3×5+9+1 = . 876204 8 × 762 + 0 + 4
3+6×0×9 3609 3×7+9+8 3798 = . 487215 (4 × (8 + 7) + 21) × 5 • 562104 = 562 × 10 + 4 . •
3807 3+8×0×7 = . 195426 1 + 9 × (5 + 4 + 2 + 6)
•
3+8×0×7 3807 = . 645921 64 × 5 + 9 × 21
•
3816 3 × 8 + 16 = . 270459 (270 + 45) × 9
•
3852 (3 + 8 + 5) × 2 = . 104967 1 + (04 + 9) × 67
•
(3 × 8 + 5) × 2 3852 = . 710694 7 + 10694
3645 3 × (6 + 4 + 5) = . 918702 9 × 18 × 70 + 2
•
3861 38 + 6 × 1 = . 279045 (2 + 7 × 90 + 4) × 5
•
3861 (3 + 8) × 6 × 1 = . 974025 9 × 74 × 025
•
3+8×7+9 3879 = . 256014 2 × (560 + 1) × 4
3×6+2+4 3624 • = . 197508 (19 + 7) × 50 + 8
•
(3 + 6) × 45 3645 = . 109872 109 × 8 × 7 × 2
3486 3+4+8+6 = . 597102 5 × (9 + 710) + 2
•
•
3502 3 + 50 × 2 = . 147968 1 × 4 × (7 + 9) × 68
•
3654 3+6+5×4 = . 109872 (1 + 09) × 87 + 2
•
3502 3×5+0+2 = . 791864 (79 + 1) × 8 × 6 + 4
•
3654 3+6+5×4 = . 978012 97 × 80 × 1 + 2
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
3897 3×8+9+7 = . 206541 20 × (65 + 41)
•
40 + 3 × 5 4035 = . 172698 17 × 2 × 69 + 8
•
4095 4×0×9+5 4329 4 + 3 × (2 + 9) = •. = . 182637 (1 + 8) × (2 + 6) × 3 + 7 567801 5 + 6 × (7 + 801)
•
3904 3 + 9 + 04 = . 127856 1 × 2 × 7 + 85 × 6
•
4035 4×0×3+5 = . 218697 2 × (1 + 86) + 97
•
4095 4×0×9+5 = . 267813 2 × 6 × 7 + 81 × 3
3906 3 + 9 + 06 • = . 154287 15 + 4 × 2 × 87
40 + 35 4035 = . • 971628 (9 × 71 + 6) × 28
4095 4×0×9+5 • = . 281736 2 × (8 × 17 + 36)
•
4362 4 × (3 + 6) × 2 = . 519078 51 × (90 + 78)
•
4392 43 + 9 × 2 = . 578016 5 + 7 + 8016
•
4503 (4 + 5 + 0) × 3 = . 927618 9 × (2 + (76 + 1) × 8)
•
3906 (3 + 90) × 6 = . 187425 (1 + 8) × 7 × 425
•
3+9+1×8 3918 4056 4 + 0 × 56 = . • = . 207654 (20 × (7 + 6) + 5) × 4 738192 7 × (3 + 8 + 1 + 92)
•
3928 (3 + 9) × 2 + 8 = . 106547 (1 + 06 × 5) × 4 × 7
•
4×0×9+5 4095 4065 4×0×6+5 = . 4509 •. 4+5+0×9 = 826371 8 × 2 × (6 + 3) × 7 + 1 • = . 173982 ((1 + 7 + 3) × 9 + 8) × 2 386271 3 + 8 × 6 × 2 × (7 + 1)
•
(3 + 9) × 2 × 8 3928 = . 601475 60 × 14 × 7 × 5
•
4105 4 × 1 × 05 4065 4 + 06 + 5 •. = . = (9 × 7 + 86) × 32 198372 (1 + 9 × 8) × (3 + 7) + 2 978632
•
3972 (3 + 97) × 2 = . 508416 50 × 8 × 4 × 16
•
4×0×6+5 4065 = . 298371 2 + 98 × 3 + 71
•
4016 4 + 0 × 16 4065 4×0×6+5 = . 935728 (9 + 3) × (5 + 72) + 8 • 381297 = (38 + 1 × 29) × 7 .
•
•
•
4038 4 × 03 + 8 = . 692517 (6 + 92) × 5 × 1 × 7
4095 4×0×9+5 = . 326781 3 + (2 + 6 × 7) × (8 + 1) 4509 4 + 5 + 09 • = . 163827 1 × 6 + 3 × 8 × 27 4095 4×0×9+5 • = . 4509 4+5+0×9 678132 6 + 7 + 813 + 2 • = . 381762 (3 + (8 + 1) × 7 × 6) × 2 •
•
(4 + 1 + 0) × 6 4106 = . 827359 82 × 73 + 59
4 + 10 × 7 4107 = . • 263958 2 × ((6 × 395) + 8)
4×0×1+8 4018 4158 (4 + 1) × 5 + 8 4 + 06 + 5 4065 = . • = . 972356 (9 + 7) × (23 × 5 + 6) • 781293 = (7 + 8 × (1 + 2)) × 93 . 207396 20 × (73 + 9) + 6 4023 4+0×2+3 = . 518967 (51 + 8 × 9 + 6) × 7
•
4072 4 × (0 × 7 + 2) = . 316598 3 + 1 + 6 × (5 + 98)
•
4158 4+1+5+8 = . 709632 (7 + 09) × 6 × 32
•
4 + 0 × 27 4027 4×0×8+5 = . 4085 539618 (5 × (3 + 9) + 6 + 1) × 8• 672391 = 6 × 72 + 391 .
•
4172 4 × (1 + 7 × 2) = . 605983 (60 + 5 × 9) × 83
•
4029 40 + 2 + 9 = . 183675 (1 + 8 × 3 + 6) × 75
•
4215 (4 + 2) × 15 = . 708963 (70 × 8 × 9 + 6) × 3
•
4092 4 + 0 × 92 4032 (4 + 03) × 2 4293 4+2+9+3 •. = = •. = . 385671 (3 × 8 + 5) × (6 + 7 × 1) 510867 158976 (1 × 5 × (8 + 9) + 7) × 6 51 × (0 × 8 + 6 × 7)
•
4086 4×0×8+6 = . 512793 51 × 2 + 7 × 93
•
4509 4+5+0×9 = . 817632 (8 + 1 + 7 × 6) × 32
•
4512 4 × (5 + 1 + 2) = . 398607 (39 + 8) × 60 + 7
•
4523 4×5×2+3 = . 809617 80 × 96 + 17
•
4536 4+5+3+6 = . 198072 1 + 9 × (80 + 7) + 2
•
(45 + 3) × 6 4536 = . 280917 28 × 091 × 7
•
4536 (4 + 5 + 3) × 6 = . 712089 (7 + 120) × 89
•
4536 4 × (5 + 3 + 6) = . 918702 9 × 18 × 70 + 2
•
4539 45 + 3 + 9 = . 267801 2 + 6 × 7 × 80 + 1
•
4 + 5 × 72 4572 = . 186309 1 + 8 × 6 × 309
4572 • = 380619 4301 4 + 30 × 1 = . • 596827 ((5 + 9) × 6 × 8 + 2) × 7 4605 • = 329718 4095 (4 + 09) × 5 4032 4×0×3+2 4+3×0×2 = . 4302 • = .• (1 + 2 × 7 × 3 × 8) × 6 • 791568 = 7 + 9 + 15 × 6 × 8 . 175896 1 × 7 × (5 × 8 + 9) + 6 127386 4607 • = 153928 4095 (4 + 09) × 5 4305 4 × 3 × 05 4032 (4 + 03) × 2 • = . • = . • = . 4608 138726 1 × 3 × (8 + 726) 819672 (8 + 1 × 9) × 672 175968 1 × 7 + 596 + 8 • = 153792
4032 4 × 03 × 2 • = . 159768 15 + 9 × (7 + 6) × 8
4092 4 + 09 × 2 • = . 758136 7 × 581 + 3 + 6
4 + (5 + 7) × 2 . (3 × (80 + 6) + 1) × 9 4+6+0×5 . 3 + 2 + 9 × (71 + 8) 4 + 6 + 07 . (1 + 5 × (3 + 9 + 2)) × 8 4 × (6 + 0 × 8) . 1 + 5 + 3 + 792
•
4032 4 × 032 = . 671895 6 × (71 + 8) × 9 × 5
•
4095 (4 + 09) × 5 = . 172368 (17 + 2) × 3 × 6 × 8
•
4032 (4 + 03) × 2 = . 715968 7 × 1 × 59 × 6 + 8
•
4608 (4 + 60) × 8 4095 4 × (0 × 9 + 5) 4308 4+3×0×8 = . = •. = .• 1 × 9 × 2375 173628 (1 + 7 × (3 + 6 × 2)) × 8 619275 (6 × 1 × 9 × 2 + 7) × 5 192375
•
4308 (4 + 3) × 08 = . 296175 2 + 9 × 61 × 7 + 5
•
4608 4+6×0×8 = . 173952 (1 + 7 + 3) × 9 + 52
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
4608 4+6+0×8 = . 571392 5 × (7 + 13 × 9) × 2
•
5012 5 + 01 + 2 = . 987364 9 × 8 × 7 × 3 + 64
•
5+1+0×3 5328 5+3+2×8 5103 = . • = . 278964 2×7×8+9×6×4 490176 (4 × 90 + 1 + 7) × 6
•
4608 (4 + 60) × 8 = . 739125 73 × 9 × 125
•
5013 5 + 0 × 13 = . 496287 (49 + 6 × 2) × 8 + 7
•
5103 5+1+0×3 = . 649782 6 × (49 + 78) + 2
•
5346 5+3+4+6 = . 197208 1 × ((9 × 7 + 20) × 8)
•
4617 4+6+1×7 = . 350892 3 × (50 × 8) + 92
•
5 + 01 + 3 5013 = . 867249 86 × (7 × 2 + 4) + 9
•
5106 5 + 106 = . 784392 7 × 84 × (3 × 9 + 2)
•
5346 (5 + 3) × 4 + 6 = . 197802 (1 + 9 × 78) × 02
•
5346 5×3×4+6 = . 918702 9 × 18 × 70 + 2
•
5376 5+3+7+6 = . 840192 8 × (401 + 9) + 2
•
(5 × 3 + 8) × 2 5382 = . 176904 1 × 7 × 6 × 9 × 04
•
5382 5 + 3 + 82 = . 904176 90 × 4 × 1 × 7 × 6
•
5401 54 + 01 = . 396728 3 × 96 × 7 × 2 + 8
•
54 + 12 5412 = . 308976 30 + 89 × 7 × 6
•
5413 5+4+1×3 = . 627908 (6 × 2 × 7 + 90) × 8
•
5418 (5 + 4 × 1) × 8 = . 273609 27 + 3609
•
5421 5+4×2×1 = . 706398 7 × (06 × 39 + 8)
5203 5+2×0×3 = . 978164 97 × 8 + 164
•
5427 54 + 2 + 7 = . 160398 (1 + 6 × 03) × 98
5208 5+2×0×8 = . 479136 4 × (79 + 1 × 36)
•
5463 (5 + 4) × 6 + 3 = . 180279 1802 + 79
5283 (5 × 2 + 8) × 3 = . 406791 4067 + 91
•
5 × (4 + 7 × 2) 5472 = . 103968 10 × 3 × (9 + 6 × 8)
4716 4 × (7 + 1) × 6 • = . 398502 39 × 8 × (50 + 2)
•
•
•
•
•
4716 4×7+1+6 = . 509328 5 × 09 × 3 × 28 4732 (4 + 7 + 3) × 2 = . 805961 8 × 0596 + 1 4739 4×7+3+9 = . 208516 20 × (8 + 5 × 16)
5+1+2×6 5126 5028 5×0×2+8 = . • • = .340879 3 × (40 × 8 + 79) 379614 ((3 + 7) × (9 + 6) + 1) × 4
•
•
•
5136 5+1+3×6 • = . 204798 20 × 47 + 9 + 8
5032 (5 + 03) × 2 = . 164798 16 + 4 + 7 × 9 × 8 5 × 03 + 2 5032 = 194768 (19 + 4 × 7) × (6 + 8)
5148 5 × (1 + 4 + 8) • = . 709632 70 × (96 + 32) .
5046 5 + 04 × 6 = . 137982 1 × 3 + 79 × (8 + 2)
4+8+0×6 4806 5 + 04 × 6 5046 = . 253917 2 × (5 + 39 × (1 + 7)) • 917328 = (9 × 1 × 73 + 2) × 8 . 4+8×0×6 4806 5 + 04 × 6 = . 5046 271539 2 + 7 × 1 × (5 + 3 × 9) • 918372 = 91 × (8 + 3 × 7) × 2 .
5168 5+1×6+8 • = . 739024 7 + 3 × 902 + 4 5 × (1 + 8) + 7 5187 = . • 309624 3096 + 2 × 4 •
• •
4806 (4 + 8 + 0) × 6 = . 732915 (7 + 3 + 2) × 915
•
5×0×4+8 5048 = . 136927 1 + 3 × 69 + 2 + 7 •
4 + 8 + 51 4851 = . • 307692 (30 + 7) × 6 × 9 × 2
5048 5 + 0 × 48 • = . 312976 (31 + 2) × 9 + 7 + 6 •
4856 4 × (8 + 5 × 6) • = . 120793 1 + 20 × 7 × 9 × 3
5049 50 + 49 • = . 182376 (1 + (82 + 3) × 7) × 6
4902 4 × 9 + 02 50 + 6 × 1 • = . 5061 317856 (3 + 1) × 7 × 8 × (5 + 6)• 428739 = 4 × 2 × (8 × 73 + 9) .
•
•
5187 5 × (1 + 8) + 7 = . 932064 9320 + 6 × 4 5 + 2 × 01 5201 = . 489637 4 + 8 × 9 × (6 + 3) + 7 5203 5 + 2 × 03 = . 194876 (1 + 9 + 48) × 7 + 6
4×9+0+2 4902 5061 5 × (06 + 1) = . 5301 5 × (30 + 1) 675831 (6 + 7) × (5 + 8) × 31 • 823497 = (82 + 3) × (4 + 9 × 7) . 5472 (5 + 4 + 7) × 2 • = . = . 289674 (2 + 8 × 96) × (7 + 4) • 309168 30 × (9 + 1) × 6 + 8 4912 4 × (9 + 1 + 2) 5072 (5 + 07) × 2 • = . = . • 5 + 3 + 01 5301 683075 (6 + 83 + 0) × 75 54 + 8 + 1 194638 1 × 9 + 4 × 6 × 38 = . 5481 • 964782 9 × (6 + (4 + 7) × 8 × 2)• 260739 = (260 + 73) × 9 . 4+9+2×1 4921 5073 5 × 07 + 3 • = . • = . 5304 5 + 3 × 04 5632 56 + 32 570836 (570 + 8) × 3 + 6 128694 (1 + 2 × 8 × (6 + 9)) × 4• = . • = . 168792 1 + 6 × 87 + 9 × 2 187904 1 × 8 × (7 + 90 × 4)
•
•
4956 4 × (9 + 5) × 6 = . 102837 (10 + 2) × 83 × 7
•
5078 5×0×7+8 = . 269134 2 × (69 + 1) × 3 + 4
•
4968 4 × 9 + 68 = . 352107 (3 + 5 × 210) × 7
•
5094 5×0×9+4 5312 (5 + 3) × (1 + 2) = •. = . 236871 2 × (3 + 6 × (8 + 7 × 1)) 687904 (687 + 90) × 4
•
4985 (4 + 9 + 8) × 5 = . 120637 (1 + 20 × 6) × 3 × 7
•
5103 5 + 1 + 03 = . 264789 2 + 64 × 7 + 8 + 9
•
•
5304 5 + 3 × 04 = . 196872 1 + 9 × 6 + 8 × 72
•
5643 56 + 4 + 3 = . 127908 12 × 7 × (9 + 08)
•
5643 (5 × 6) + 43 = . 208791 20 × (8 + 7) × 9 + 1
5716 5+7+1×6 5328 5 × 32 + 8 = . = . • 302948 3 × 02 + 948 190476 (1 + 90) × (4 + 7) × 6
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
5731 5 × (7 + 3 + 1) = . 420968 42 × 096 + 8
•
5934 (5 + 9) × 3 + 4 = . 210786 (2 + 10 + 7) × 86
•
6048 6 + 0 × 48 6291 6+2×9×1 = •. = . 397152 (3 × (9 × 7 + 1) + 5) × 2 807345 8 × (073 + 4) × 5
•
5814 58 + 1 + 4 = . 720936 7 × 2 × 093 × 6
•
5943 (59 + 4) × 3 = . 108672 1 × 08 × 6 × 72
•
6314 6+3+1+4 6048 6×0×4+8 = . • 890725 = (8 × 9 + 0 + 7) × 25 . 715932 7×1×5×9×3+2
•
5832 5+8+3×2 = . 104976 (1 + 049 + 7) × 6
•
(5 + 9) × (6 + 4) 5964 = . 130782 1 × 307 × (8 + 2)
•
6052 6×0×5+2 = . 947138 9 + 4 × (71 + 3) + 8
•
6318 6 × 3 + 18 = . 527904 (5 + 27) × (90 + 4)
6072 6+0×7+2 • = . 314985 3 + 1 × 4 × (98 + 5)
•
63 × 2 + 7 6327 = . 104895 (1 + 048) × 9 × 5
•
6327 6+3×2+7 = . 105894 10 + (5 + 8 × 9) × 4
•
6327 6+3×2+7 = . 140859 1 + 408 + 5 + 9
•
6327 6+3×2+7 = . 194805 (1 + 9 × 4 + 80) × 5
•
6+3×2+7 6327 = . 491508 4 × 9 × (1 + 5 × 08)
•
6342 6 + 34 + 2 = . 197508 (19 + 7) × 50 + 8
5832 5 + (8 + 3) × 2 • = . 147096 (1 + 4 + 70) × 9 + 6
•
•
5 × 8 + 32 5832 = . 714906 7 × 14 × 90 + 6 5841 5 × (8 + 4 × 1) = . 327096 32 × 7 × (09 + 6)
5967 (5 × 9 + 6) × 7 • = . 123084 (1 + 230 × 8) × 4
•
•
5967 5 + 96 + 7 = . 143208 1 × (4 + 320) × 8 59 + 6 + 7 5967 = . 310284 (310 + 2) × (8 + 4)
•
•
6 × (0 × 7 + 2) 6072 = . 581394 5 + 8 × (139 + 4) 6084 6 × 08 + 4 = . 132795 1 + 3 × 27 × (9 + 5)
58 × 6 × 2 5862 = . 794301 7 + 94301
•
5892 5+8+9+2 = . 643701 6 × 437 × 01
•
•
5901 5 + 9 × 01 = . 287463 2 × 8 + 74 × (6 + 3)
•
6012 6 × 01 × 2 6 × 1 × 02 6102 6 + 34 + 5 = .• = . 6345 = . 978453 9 × 7 × (8 + 4 × 5 + 3) 978354 (9 + 7) × 8 × 3 × 5 + 4 • 120978 1 × (2 + 09) × 78
•
5 + 9 × 04 5904 = . 128736 (128 + 7 × 3) × 6
•
6129 6+1×2×9 6015 6×0×1+5 •. = . = 5 + 7 × 40 × 8 + 3 473982 4 × ((7 + 3) × 9 + 8) + 2 574083
•
6172 6 + (1 + 7) × 2 5904 (5 + 9) × 04 6027 6×0×2+7 = . = . • = . • 549308 5 × (4 + 9) × 30 + 8 137268 (13 × 7 + 2) × (6 + 8) 148953 14 + ((8 + 9 × 5) × 3)
•
(5 + 9) × 04 5904 = . 176382 1 + 76 × (3 + 8) × 2
•
•
5904 5 + 9 × 04 • = . 237168 23 × 71 + 6 + 8 5904 5 + 9 + 04 • = . 386712 3 + (8 + 6) × 7 × 12
•
•
•
5907 59 + 07 = . 163248 1 × ((6 + 32) × 48) 5907 5 + 9 + 07 = . 614328 6 × (1 + 4 × 3) × 28 5913 5+9+1+3 = . 270684 2 × 70 + 684
•
5973 59 + 73 = . 106428 (1 + 06) × 42 × 8
•
6012 6 × 01 × 2 = . 743985 74 × (3 + 9 + 8) + 5
•
6084 6 × 08 + 4 = . 213759 21 × (3 + 75 + 9) 6084 6 × 08 + 4 = . 921375 (92 + 13) × 75
6192 (6 + 1 + 9) × 2 6027 6×0×2+7 = . = . • 538704 (5 + 3) × 87 × 04 458913 (45 + 8) × (9 + 1) + 3
•
6351 6 × (3 + 5 × 1) = . 489027 48 × (9 + 02) × 7
•
6357 (6 + 3) × 5 + 7 = . 982401 98 × 2 × (40 + 1)
•
6358 6+3+5+8 = . 149702 (1 + 4 × 9) × 7 × 02
6372 • 6201 6 × 2 + 01 6032 6 + 03 × 2 450819 = . • = .• (4 + 7 × 9 × 3 + 8) × 5 197548 1 × 9 + (7 + 5) × 4 × 8 479385 6384 • 105792 6 × 2 + 09 6209 6039 6 + 03 × 9 = . • = . • 137485 (1 + (3 × 7 × 4) + 8) × 5 178425 1 × ((7 + 8 × 4) × 25) 6384 • 109725 6237 (6 + 2 + 3) × 7 6045 60 + 4 × 5 = . = . • • 159084 (1 + 5 × (90 + 8)) × 4 178932 (1 + 7) × (8 + 9 × 32) 6402 • 378591 6237 (6 + 2 + 3) × 7 6048 6 × 04 + 8 = . • = . • 159408 (1 + 5 × (9 + 40)) × 8 159327 1 + 5 + 93 × (2 + 7) 6402 • 579381 6237 6×2+3×7 6048 6 × 048 • = . • = . 6408 185409 18 × 54 + 09 291375 (2 × 91 + 3) × 75 • 127359
•
5926 5 + 9 × (2 + 6) = . 314078 3 + 1 × 4078
•
6048 6 + 04 + 8 = . 351792 3 + (51 + 7) × 9 × 2
•
•
5928 (5 + 9 + 2) × 8 = . 603174 60 × 31 × 7 + 4
•
6048 6×0×4+8 = . 371952 3 × (7 + 1) + 9 × 52
•
=
6 × (3 + 7 + 2) . 4 + 5081 + 9
=
6 × (3 + 8) + 4 . (1 + 0579) × 2
=
(6 + 3) × 8 × 4 . 10 × (97 + 2) × 5
=
64 + 02 . 3 + 78 × 5 × (9 + 1)
=
6+4+0×2 . 5 × (7 + 93 + 81)
=
6 × (4 + 0 × 8) . (1 + 2 + (7 + 3) × 5) × 9
6237 (6 + 2 + 3) × 7 = . 491508 4 × (9 + 1508)
•
6408 6 × (40 + 8) = . 732915 732 × 9 × 1 × 5
6 × (2 + 8 + 9) 6289 = . 130745 1 × 30 × (74 + 5)
•
6415 6+4+1×5 = . 790328 (7 × 9 + 0 + 3) × 28
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
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6435 (6 + 4) × (3 + 5) = . 120978 (1 + 20 × 9 + 7) × 8
•
6804 6+8+0×4 = . 375192 3 + 751 + 9 × 2
•
7065 7×0×6+5 (7 + 1) × 4 × 6 7146 = .• = . 189342 1 × 8 + 9 × (3 + 4) × 2 803925 80 × 3 × 9 × 2 × 5
•
6489 6+4+8+9 = . 253071 2 + 5 × 30 × 7 + 1
•
6804 6+8+0×4 = . 579312 5 × (79 × 3 + 1) + 2
•
7065 7×0×6+5 = . 298143 2 × 98 + (1 + 4) × 3
•
7182 7+1+8×2 = . 436905 (4 + 3 × (6 + 90)) × 5
•
6501 6 + 5 × 01 = . 349872 3 + 4 + 9 + 8 × 72
•
6912 (6 + 9 + 1) × 2 = . 350784 (350 + 7 × 8) × 4
•
7×0×6+5 7065 = . 392814 3 × (9 + 2) × 8 + 14
•
7194 7+1+9×4 = . 238056 (2 + 3 × 8) × 056
7068 7×0×6+8 • = . 321594 3 × 2 × (1 + 59) + 4
•
(7 + 2) × 09 7209 = . 465381 4 + 653 × 8 + 1
•
7209 7+2+0×9 = . 531864 (5 × (3 + 1) × 8 + 6) × 4
•
(7 + 2 × 1) × 5 7215 = . 369408 (3 + 69) × 4 × 08
•
7238 7 + (2 × 3 + 8) = . 150964 (1 + 5) × (09 + 64)
•
7 + 2 + 38 7238 = . 196504 (19 + 6 × 50) × 4
•
7 × (25 + 9) 7259 = . 106384 (106 + 3) × 8 × 4
•
7302 7+3+0×2 = . 419865 (4 + 1) × (9 + 8 + 6) × 5
•
7308 7 × 3 + 08 = . 125496 (1 + 2 × (5 + 4 × 9)) × 6
•
7308 7+3×0×8 = . 164952 (1 + 64 + 9 + 5) × 2
•
7308 7×3+0+8 = . 641592 6 × (415 + 9) + 2
6528 6 + 5 × (2 + 8) • = . 137904 13 × (7 × (9 + 04))
•
6+5+3×1 6531 = . 472098 4 + 7 × 2 × 09 × 8 65314 (65 + 3) × 14 = . 90287 ((90 × 2) + 8) × 7
(6 + 9 + 1) × 2 6912 = . • 738504 7 + (3 + 850) × 4
•
6912 6+9×1×2 = 753408 (7 + (5 + 3) × 40) × 8
7068 (7 + 06) × 8 • = . . 432915 (4 + 3) × 2 × 91 × 5
6942 (6 + 94) × 2 = . 513708 5 × 1 × 370 × 8
•
6 + 97 + 2 6972 = . 103584 10 × 3 × (5 + 8) × 4
•
6702 6 + 7 + 02 • = . 395418 3 + (9 × 5 + 4) × 18
7032 7 + 03 + 2 • = . 198654 (19 + 8 × 6) × 5 + 4
•
6723 6+7+2+3 • = . 184509 1 × 8 + (4 + 50) × 9
7 × 03 × 5 7035 = . • 264918 26 + 491 × 8
•
•
•
6534 6+5+3+4 = . 918027 9 + 180 × 2 × 7
6723 6 × (7 + 2) + 3 = . 584901 58 + 4901
•
•
•
7035 7 × (0 × 3 + 5) = . 289641 2 × 8 × 9 × (6 + 4) + 1
•
•
• •
6724 6+7×2+4 = . 310985 3 × (10 + 9 × 8 × 5)
•
7035 7 × (0 × 3 + 5) = . 421698 4 + (2 + 1) × 698 •
•
6725 6+7×2+5 7 + 03 + 5 7035 = . • = . 109483 (1 + 09 × 4) × (8 + 3) 621894 (6 × 2 + 1) × (8 + 94) •
6739 6 × (7 + 3) + 9 • = . 518024 51 × (80 + 24) 6783 6 + (7 + 8) × 3 • = . 201495 20 + 1495
•
•
(6 + 8) × 04 6804 = . 137295 (1 + 3 × 72 + 9) × 5 6804 6 + 8 × 04 = . 139725 1 × (39 + 7) × 25
7 + 0 × 84 7084 = . 129536 1 × 2 + 9 × (5 + 3 + 6) 7086 7×0×8+6 = . 152349 1×5×2×3×4+9 7091 7 + 0 × 91 = . 284653 2 + 84 + 65 × 3 7×0×9+2 7092 = . 145386 1 × (4 + 5) × 3 + 8 + 6 7092 7×0×9+2 = . 386514 3 + 86 + 5 × 1 × 4 7×0×9+2 7092 = . 563814 5 × (6 + 3 × 8 + 1) + 4 7104 7 + 1 + 04 = . 285936 2 + 8 × 59 + 3 + 6 (7 + 1) × 04 7104 = . 369852 3 × 69 × 8 + 5 × 2
7308 • = 7035 7×0×3+5 912456 • = . 862491 (8 × (6 + 2) + 4) × 9 + 1 7104 (7 + 1 + 0) × 4 • = . 7326 589632 (5 + 8 × 9 + 6) × 32 • = 105894 7036 7 + 03 + 6 • = . 214598 2 × 1 × (4 × 59 + 8) 7106 7 × 10 + 6 • = . 7326 342958 (3 + 4) × (2 + 9 × 58) • = 140859 7038 7 + 03 + 8 = . • 7128 7 + 1 + 28 214659 21 × 4 × 6 + 5 × 9 7326 • = . • = 340956 3 × (4 + 095 × 6) 194805 •
7 × (0 × 4 + 2) 7042 = . (7 + 1 × 2) × 8 395861 3 + (9 + 5) × 8 × (6 + 1) 7128 • = . 436095 4360 + 9 × 5
7 × (3 + 0 × 8) . (9 × 12 × 4 + 5) × 6 7+3+2×6 . 10 + (5 + 8 × 9) × 4 7+3+2×6 . 1 + 408 + 5 + 9 7+3+2×6 . (1 + 9 × 4 + 80) × 5
•
7326 7+3+2×6 = . 491508 4 × 9 × (1 + 5 × 08)
•
6804 6×8+0×4 7042 7 + 0 × 42 = . • = . 172935 (1 + (72 + 9) × 3) × 5 536198 5 + (3 + (6 + 1) × 9) × 8• 7128 = 7 + 1 + 2 + 8 . 593604 59 + 360 × 4
•
7348 7×3+4+8 = . 215096 (2 + 150 + 9) × 6
•
6804 6 + 8 + 04 = . 175392 1 + 7 × 53 + 92
•
7046 7 + (0 × 4 + 6) = . 129538 1 + 2 × (95 + 3 × 8)
•
7352 7+3+5×2 = . 406198 (4 + 061) × (9 + 8)
•
6804 6 × (80 + 4) = . 192375 19 × 2 × 375
•
7392 7056 7×0×5+6 7+3+9×2 7+1×3+2 = . • 7132 = = . . • 984312 9 × (84 + 3 × (1 + 2)) 105864 1 + 05 × 8 × (6 + 4) 859406 8×5×9×4+0+6
•
7128 7+1+2×8 = . 634095 6 × 340 + 95
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
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7395 (7 + 3 × 9) × 5 = . 164082 1 × (6 + 40) × 82
•
7614 7+6+1+4 = . 208539 2 × 08 + 53 × 9
•
8091 8 + 09 × 1 8235 (8 + 2 + 3) × 5 = . • = . 752463 (7 + 52 × (4 + 6)) × 3 760914 (7 × 60 + 9) × 14
•
7408 7+4×0×8 = . 659312 65 + 9 × 31 × 2
•
7614 7 × (6 + 1 × 4) = . 209385 20 × (9 + 3) × 8 + 5
•
8094 8 × 09 + 4 = . 126735 (1 + 26 + 7) × 35
•
8253 8+2×5+3 = . 109647 1 × 09 × (6 × 4 + 7)
•
7416 7 + 4 × (1 + 6) = . 830592 8 × (30 + 5 × 92)
•
7614 7+6+1+4 = . 305829 3 + 05 × 8 × 2 × 9
•
8096 80 + 96 = . 217534 21 × 75 × 3 + 4
•
8 × (2 + 6 + 4) 8264 = . 391507 3 × (9 + 1507)
•
(7 + 4 × 2) × 8 7428 = . 109563 10 × (9 + 56 × 3)
7+6+1+4 7614 = . • 829503 (8 + 29) × (50 + 3)
7452 7+4+5+2 • = . 319608 3 + 1 + 96 × 08 7452 7+4+5×2 • = . 961308 9 × (61 + 30 × 8) 7483 7+4×8+3 • = . 906512 (90 + 6) × (51 + 2) 7+5×0×4 7504 = . • 638912 6 × (3 + 8) × 9 × 1 + 2
•
•
•
•
•
•
•
7506 7+5+0×6 = . 238941 2 + 3 + 8 + 9 × 41
•
•
•
•
7653 = 290814 7853 = 416209
=
8 × 2 + 74 . (1 × 5 + 30 × 6) × 9
=
8×3+0×4 . (2 × 7 × 1 × 9 + 5) × 6
=
(8 + 3 + 5) × 2 . (1 × 7 + 4) × 60 + 9
=
(8 + 3 + 5) × 2 . (1 + 9) × (4 + 70) + 6
=
8 + 3 × (5 + 2) . 1 × 9 + 670 + 4
•
8109 8 + 1 × 09 = . 8352 (8 + 3 + 5) × 2 = . 423576 4 × 2 × (3 × 5 × 7 + 6) • 401679 (4 + 0167) × 9
•
7946 (7 + 9 + 4) × 6 = . 135082 (1 + 3) × (508 + 2)
•
8109 8 + 1 × 09 = . 476523 476 + 523
•
8+3×5+7 8357 = . 142069 (14 + 20) × (6 + 9)
•
7 × 9 + 56 7956 = . 104832 (1 + 048) × 32
•
8109 8 + 1 × 09 = . 675432 6 × (75 + 43) × 2
•
8372 83 + 7 + 2 = . 519064 5 × 190 × 6 + 4
•
8019 80 + 19 = . 374625 (3 + 7) × 462 + 5
•
8135 (8 + 1 + 3) × 5 = . 240796 240 × 7 + 96
•
8379 8+3×7+9 = . 265041 2 + 6 × 50 × 4 × 1
7+5+0×6 7506 = . 918234 91 × 8 × 2 + 3 × 4 7512 7 + 5 + 12 = . 640398 6 + 40 × 3 × (9 + 8)
8019 80 + 19 • = . 437562 4 × 3 × 75 × 6 + 2
8394 (8 + 3 × 9) × 4 • = . 8136 8×1×3+6 256017 2 × 5 × (60 + 1) × 7 • = . 927504 9 × (2 + 7 × (50 + 4))
8×0×4+6 8046 = . • 391572 3 × 91 + 5 + 7 × 2
81 + 5 + 4 8154 = . • 703962 70 × (3 + 9 × 6 × 2)
8046 8×0×4+6 • = . 7602 7×6+0×2 792531 7 + (9 + 2) × 53 + 1 • = . 143895 (1 + 4) × 3 × (8 + 9 × 5) 7605 (7 + 6) × 05 • = . 128934 12 × 89 + 34
8+2×6+9 . 14 × 05 × 7 + 3
7902 (7 + 9) × 02 = . 458316 4 × (5 + 8 × 3) × 16
7506 7+5+0×6 = . 893214 (8 + 9) × 3 × 2 × 14
7518 (7 + 5 × 1) × 8 = . 962304 96 × (2 + 30) × 4
7641 = 290358
=
•
7506 7+5+0×6 = . 392814 39 × 2 × 8 × 1 + 4
7 + 5 + 14 7514 = . 608923 60 + 89 × 23
7635 = 248901
8269 • 8 + 1 × 06 8106 140573 = . • 379245 (3 + (7 + 9) × 2 × 4) × 5 8274 • 153069 7+6×3+5 8109 8 + 1 × 09 . • = . 2 × 489 × 01 237546 (2 + 3 × (7 + 5 × 4)) × 6 8304 • 271956 7+6+4×1 8109 8 + 1 × 09 . • = . 2 × (9 × 035 + 8) 253764 (2 + 5 + 3 × 7 × 6) × 4 8352 • 174609 7 × 6 + 53 8 + 1 × 09 . • 8109 = . (2 + 90 × 8) × (1 + 4) 264735 (26 + 4 + 7) × 3 × 5 8352 • 194706 78 + 5 × 3 8109 8 + 1 × 09 . • = . 8352 41 × 6 × 20 + 9 326745 3 × 2 + 674 + 5 • 196704
•
•
8064 8 + 06 × 4 = . 137592 13 × (7 + 5 + 9) × 2
•
•
8395 8+3×9+5 = . 127604 1 × 2 × 76 × 04
•
8415 (8 × 4 + 1) × 5 = . 203796 2 × 037 × 9 × 6
•
8415 8 × (4 + 1) + 5 = . 270963 (2 × 7 + 09) × 63
•
8436 8 + 4 × 36 = . 195027 ((1 + 9) × 50 + 2) × 7
8175 (8 + 1 × 7) × 5 = . 236094 2 × 3 + 60 × 9 × 4 8204 8 + 2 + 04 = . 571936 5 × (7 + 1) + 936
8205 8 + 2 + 05 = . 173946 (1 + 7) × 3 × (9 + 4) + 6• 8451 = 8 × (4 + 5 × 1) . 729603 7 × 296 × 03
•
7605 (7 + 6) × 05 = . 184392 1 × 8 + 4 × 392
•
8064 8 + 06 × 4 = . 153972 1 × 539 + 72
•
•
7605 (7 + 6) × 05 = . 198432 1 + 9 + 843 × 2
•
8073 8 + 07 + 3 = . 952614 9 × (52 + 6 + 1) × 4
•
8206 8 × 2 + 06 = . 149573 1 × 4 × 95 + 7 × 3
•
8476 8 + 4 × (7 + 6) = . 392015 3 × (920 + 1 × 5)
•
7605 (7 + 6) × 05 = . 419328 (4 + 1 + 9) × 32 × 8
•
8076 8 × (07 + 6) = . 312945 31 × 2 × (9 + 4) × 5
•
8×2×1×6 8216 = . 574093 (5 + 740) × 9 + 3
•
8502 (8 + 5) × 02 = . 173964 1 × 7 × (3 + 9 + 64)
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
8502 (8 + 5) × 02 8934 8 × 9 + 34 = .• = . 496713 4 × (9 × 6 × 7 + 1) + 3 527106 (52 + 7) × 106
•
8512 8 × (5 + 1) × 2 = . 396074 3 × 9 + 60 × 74
8+6×0×4 8604 = . • 359217 35 × 9 + 2 + 17
•
8612 (8 + 6 × 1) × 2 = . 437059 4 × (3 + 70 × 5) + 9
•
8934 8 + 93 + 4 = . 607512 60 × 7 × (5 + 12)
8937 8 + 93 + 7 • = . 104265 1 × 042 × 6 × 5 •
8937 8 + 9 + 37 = . 402165 (40 × 2 + 1) × 6 × 5
•
9078 9×0×7+8 9234 9 + 2 + 34 = •. = . 154326 1 × (5 + 4 × 3) × (2 + 6) 580716 5 × (80 × 7 × 1 + 6)
•
9102 (9 + 10) × 2 = . 468753 4 × 68 × 7 + 53
•
•
8639 8 + 6 + 39 = . 107254 10 + 72 × (5 + 4)
•
8961 (8 + 9) × 6 + 1 = . 470235 47 × 023 × 5
•
8649 8 + 6 × 49 = . 302715 302 × 7 × 1 × 5
•
9015 9×0×1+5 (9 + 1) × 3 + 5 = . 9135 472386 4 + (7 × (2 + 3) + 8) × 6• 647802 = (6 × 4 + 7) × 80 + 2 .
•
8694 (8 + 6) × (9 + 4) = . 352107 (3 + 5 × 210) × 7
•
9 + 01 + 8 9018 = . 572643 57 × 2 × (6 + 4) + 3
8946 8+9+4×6 = . 107352 (1 + 07 × 35) × 2
•
•
9×0×3+5 9035 8703 8 + 7 + 03 = . • = . • 218647 2 × (1 + (8 + 6) × 4) + 7• 461259 (4 + 6 × (12 + 5)) × 9
•
•
8704 8 × 7 × 04 = . 125936 1 + 2 × 5 × 9 × 36
•
9042 9+0×4+2 = . 163578 16 + 3 × (5 + 7 × 8)
9234 9×2+3×4 = . 761805 (7 + 61 × (8 + 0)) × 5
9261 • 9 + 1 + 05 9105 430857 = . • 367842 (3 + (67 + 8) × 4) × 2 9261 • 435708 9105 9 + 1 + 05 • = . 826734 8 + 2 × (673 + 4) 9261 • 857304 9126 (9 + 1) × 2 + 6 • = . 9261 504738 50 × 4 × 7 + 38 • 870534
8631 8 × 6 + 31 = . 794052 79 × (40 + 52)
•
•
•
9045 9 × (0 × 4 + 5) 8706 8+7×0×6 •. = . = (1 + (27 + 8) × 3) × 6 • 239415 (2 + 3 × (9 + 4 + 1)) × 5 127836
9132 (9 + 13) × 2 = . 764805 (7 × 6 + 4) × 80 + 5
=
9+2×6×1 . 4 × 30 + 857
=
9+2×6×1 . 4 × 35 × 7 + 08
=
9+2×6×1 . 8 × (5 + 7 × (30 + 4))
=
9×2+6×1 . 8 × (70 + 53 × 4)
•
9264 9×2×6×4 = . 105378 (10 + 53) × 78
•
(9 + 2 × 6) × 8 9268 = . 130745 1 × 30 × (74 + 5)
9305 • (9 + 1) × 3 + 5 9135 124687 = . 670248 6 × (70 × (2 + 4) + 8) 9306 • 172584 9152 (91 + 5) × 2 = . 640783 6 × 40 × 7 × 8 + 3 9308 • 251674 9 + 153 9153 = . 842076 (8 + 4) × 207 × 6 9308 • 514267 9156 (9 + 1) × 5 + 6 = . 9312 740328 (7 × 40 + 3) × 2 × 8 • 856704
=
930 + 5 . 1 + 24 × 6 × 87
=
9 × 3 + 06 . (1 + (7 × 2 + 5) × 8) × 4
=
(9 + 30) × 8 . 2 × (51 + 6) × 74
=
(9 + 3 + 0) × 8 . 51 × 4 × 2 × (6 + 7)
=
9+3+1×2 . (8 × 5 + 6) × 7 × 04
•
8712 8+7×1×2 = . 395604 395 + 604
•
9 × (0 × 4 + 5) 9045 = . 168237 1 + 6 + 823 + 7
•
8712 8+7×1×2 = . 593604 59 + 360 × 4
•
9045 9×0×4+5 9348 93 × 4 + 8 9+2+0×7 = . 9207 186327 1 × 8 × (6 + 3 × 2) + 7 • 145638 = (1 + 4) × 5 × 6 + 3 × 8 . • 107256 = 10 + 725 × 6 .
•
8 + 7 + 21 8721 = . 459306 (4 + 59) × 30 + 6
•
(9 + 04) × 8 9048 = . 176523 1 + (7 + 6) × 52 × 3
8749 (8 + 7) × 4 × 9 • = . 306215 30 × 6 × 21 × 5
•
•
8769 8+7+6+9 = . 254301 2 × (5 + 430 × 1)
•
•
9165 9+1×6+5 = . 304278 304 × 2 + 7 × 8
9072 (9 + 07) × 2 = . 146853 1 + 4 + 6 × 85 + 3
•
9072 9 × 07 × 2 8904 8+9×0×4 •. = . = 163584 (16 × 35 + 8) × 4 361725 ((3 + 6 × 1) × 7 + 2) × 5
•
9342 (9 + 3 × 4) × 2 = . 580761 5 × (80 + 7) × 6 + 1
•
9348 93 × 4 + 8 = . 107625 1 × 07 × 625
•
9376 (9 × (3 + 7)) + 6 = . 518024 51 × (80 + 24)
•
9406 9+4×0×6 = . 583172 (58 + 3 + 1) × (7 + 2)
•
9408 9 × (4 + 0 × 8) = . 512736 (51 × 2 + 7) × 3 × 6
9216 9 × (2 + 1 × 6) = . 307584 3 + 075 × 8 × 4
•
9423 9×4+2×3 = . 650187 (6 + (50 + 1) × 8) × 7
•
9436 94 + 3 × 6 = . 102785 10 × 2 × (7 × 8 + 5)
•
9478 (9 + 47) × 8 = . 356102 3 × 5610 + 2
9207 9+2+0×7 = . 345681 345 + 68 × 1
9072 9 + 07 + 2 9+2+0×7 • = . 9207 134568 1 + (3 + 4) × (5 × 6 + 8)• 614358 = 614 + 3 × 5 × 8 . •
•
9207 (9 + 2 + 0) × 7 = . 813564 81 × (3 × 5 + 6) × 4
•
8904 8 + 9 + 04 = . 376512 37 × (6 + 5 + 1) × 2
•
9072 9 + 0 × 72 = . 183456 18 × (3 + 4) + 56
•
9234 9 × (2 + 34) = . 106875 10 × (68 + 7) × 5
•
8924 8+9+2+4 = . 130756 1 × 307 + 5 × 6
•
(9 + 07) × 2 9072 = . 651483 6 × (51 + 4 × 83)
•
9234 9 + 2 + 34 = . 108756 10 × 8 + 75 × 6
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
9512 9+5+1×2 = . 803764 8 + 03 × 7 × 64
•
96 + 03 9603 = . 127458 1274 + 5 × 8
•
9713 9 + (7 + 1) × 3 = . 204856 20 × 4 × 8 + 56
•
9534 9+5+3×4 = . 162078 1 × 62 × 07 + 8
•
9612 9×6×1×2 = . 547083 5 + (4 + 70) × 83
•
9732 9 + 7 + 32 = . 150846 (15 × 08 + 4) × 6
9534 (95 + 3) × 4 = . 286701 28 × (6 × 70 + 1)
•
9672 9+6+7+2 = . 135408 1 + 3 × 5 + 40 × 8
•
•
•
9576 (9 + 5) × 76 = . 104832 (10 + 4) × 832
•
•
9576 9+5+7×6 = . 123804 (1 + 2) × 3 × 80 + 4
97 + 02 9702 = . • 183456 1 × 8 × (34 + 5) × 6
•
9576 9+5+7×6 = . 213408 2 × 13 × (40 + 8)
•
2.4
9702 9×7+0×2 = . 153846 153 + 846
9702 9×7+0×2 = . 354816 (3 + 5) × 48 × 1 × 6
•
9834 (9 + 8 × 3) × 4 = . 265071 2 + 6 + 50 × 71
•
9 × (8 + 4 + 1) 9841 = . 276305 (27 + 630) × 5
•
9846 98 + 46 = . 172305 1 × 72 × (30 + 5)
•
9846 9 + (8 + 4) × 6 = . 315072 (31 + 5) × 072
•
9856 98 + 5 × 6 = . 241703 (2 + 41) × (70 + 3)
9735 9 × (7 + 3) × 5 = . 124608 (1 + 2) × 4 × 60 × 8 9735 9 × (7 + 3 + 5) = . 218064 21 × (80 + 64)
9756 (9 × 7 + 5) × 6 • = . 102438 102 × (4 + 38)
•
•
9 × (8 + 0) + 4 9804 = . 623715 (6 × 23 × 7 + 1) × 5
Fractions Ending in Zero in Numerator
The above subsection 2.3 brings results having 0 in between each fraction. This subsection brings fractions ending in 0 in the numerator, but dividing by 10 don’t guarantee the existence of selfie fraction as in case of subsection 2.1.
•
190 1+9+0 = . 342 3 × (4 + 2)
•
190 1+9+0 = . 6745 (67 + 4) × 5
•
460 4×6+0 = . 2875 2 × (8 + 7) × 5
•
280 2×8+0 = . 315 3 + 15
210 2 + 10 • = . 4375 (43 + 7) × 5
•
480 4×8+0 = . 1365 1+3×6×5
280 2×8+0 • = . 735 7 + 35 280 2×8+0 • = . 945 9 + 45
•
•
510 5 + 10 • = . 782 7+8×2 • 540 5×4+0 • = . 837 8×3+7 140 1×4+0 = . 2695 2 + (6 + 9) × 5 140 1×4+0 = . 2765 2 + 7 × (6 + 5)
140 1×4+0 = . • 8295 8 × 29 + 5
230 2+3+0 = . 5796 (5 + 7 + 9) × 6
540 5 + 40 • = . 1296 1×2×9×6
280 2×8+0 = . 1365 13 + 65 •
• •
2+3+0 230 = . 1564 1×5×6+4
5×2+0 520 = . • 4836 4 + 83 + 6
• •
•
210 2×1+0 = . 9765 9×7+6×5
480 4×8+0 • = . 6915 6 + 91 × 5
280 2×8+0 = . 4375 (43 + 7) × 5 •
•
280 2+8+0 = . 4536 (4 + 5) × 3 × 6
540 5×4+0 = . 1782 (1 + 7) × 8 + 2 540 5×4+0 = . 2673 26 + 73 540 5×4+0 = . 2916 2×9×1×6
•
580 5×8+0 = . 2146 2 + 146
•
580 5×8+0 = . 3219 3 + 219
•
580 5×8+0 = . 4176 (41 + 7) × 6
•
620 6+2+0 = . 3875 3 × (8 + 7) + 5
•
6×3+0 630 = . 1925 1 × (9 + 2) × 5
•
630 6×3+0 = . 8295 8 × 29 + 5
•
810 8 + 10 = . 2745 2×7×4+5
•
810 8 + 10 = . 3465 3×4×6+5
•
810 8 + 10 = . 6345 6 + 3 × 45
•
540 5×4+0 = . 3726 (3 × 7 + 2) × 6
•
3+2+0 320 = . 7168 (7 + 1 + 6) × 8
870 8+7+0 = . 1392 (1 × 3 + 9) × 2
•
540 5×4+0 = . 6318 6 × (31 + 8)
•
870 8+7+0 = . 3596 3+5+9×6
•
580 5×8+0 = . 1276 12 + 76
•
870 8 + 70 = . 3625 (3 + 62) × 5
•
580 5×8+0 = . 1392 1 + 3 + 92
•
9+1+0 910 = . 4732 4 × (7 + 3 × 2)
•
180 1×8+0 = . 2475 2 × (4 + 7) × 5
•
•
190 1+9+0 = . 3857 (3 × 8 + 5) × 7
•
3×4+0 340 = . 1275 (1 × 2 + 7) × 5
•
190 1+9+0 = . 6574 6 × 57 + 4
•
4+1+0 410 = . 3526 3 + 5 × (2 + 6)
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
960 9+6+0 = . 1472 1 + (4 + 7) × 2
•
1540 15 × 4 + 0 = . 6237 6 + 237
•
2490 2 + 4 + 90 = . 8715 8 × 7 × (1 + 5)
•
960 9+6+0 = . 4352 4 × (3 × 5 + 2)
•
1540 1×5×4+0 = . 7623 76 + 23
•
1240 12 + 4 + 0 • = . 3875 3 × (8 + 7) + 5
•
1560 1 × 5 + 60 = . 2784 2×7×8+4
•
•
1260 12 + 60 = . 4375 (43 + 7) × 5 1×2+8+0 1280 = . 4736 4×7+3+6
15 + 70 1570 = . • 4396 4 + 39 × 6 1580 1×5×8+0 • = . 3476 3 × 4 + 76 •
•
1290 1 + 29 + 0 = . 4386 4×3×8+6 •
•
1 + 3 + 20 1320 = . 4895 4 + (8 + 9) × 5 •
•
13 + 2 + 0 1320 = . 5896 5+8+9×6 •
1380 1×3×8+0 • = . 2645 26 + 4 × 5 • •
1380 1×3×8+0 = . 7245 7 × 2 × (4 + 5) 1420 (1 + 4) × 2 + 0 = . 5396 5+3×9+6
•
14 + 6 + 0 1460 = . 3285 3 × (2 + 8 + 5)
•
1460 1×4+6+0 = . 9782 9+7×8+2
•
1470 1 + 4 + 70 = . 3528 3 × (52 + 8)
1470 1×4×7+0 • = . 8925 (8 + 9) × 2 × 5
•
•
1480 (1 + 4) × 8 + 0 = . 3256 32 + 56 1490 1 + 4 + 90 = . 3576 (3 + 5 × 7) × 6
3420 34 + 2 + 0 = . 9785 9×7+8×5
2640 2 + (6 + 40) = . 9185 9 × 18 + 5
•
3520 3 + 52 + 0 = . 7168 (7 + 1 + 6) × 8
2×6+4+0 2640 = . • 9735 9 + (7 + 3) × 5
•
3570 3×5×7+0 = . 4692 46 + 92
•
3570 3×5×7+0 = . 4896 48 + 96
•
3690 3 × (6 + 9) + 0 = . 4182 41 + 8 + 2
2730 27 + 3 + 0 = . 4186 (4 + 1) × 8 + 6
•
3690 3 × (6 + 9) + 0 = . 5248 52 + 4 + 8
•
2830 (2 + 8) × 3 + 0 = . 7641 76 + 4 + 1
•
3690 3 × (6 + 9) + 0 = . 5412 54 + 12
•
2×8+4+0 2840 = . 5396 5+3×9+6
•
3 + 7 + 20 3720 = . 9548 9×5+4×8
•
2860 (2 + 8) × 6 + 0 = . 4719 (4 + 7 × 1) × 9
•
3870 3 × (8 + 7) + 0 = . 6192 6 × (1 + 9 + 2)
•
3 + 9 + 20 3920 = . 4165 4+1×6×5
•
3960 3 × (9 + 6) + 0 = . 4752 47 + 5 + 2
•
3960 3 × (9 + 6) + 0 = . 7128 71 + 2 + 8
2970 2×9+7+0 = . 5346 5 + 34 + 6
•
3960 3 × (9 + 6) + 0 = . 8712 87 + 12
•
2+9+7+0 2970 = . 6435 6×4+3×5
•
3+9×8+0 3980 = . 7164 71 + 64
•
2980 2 + 98 + 0 = . 5364 5 × (3 + 6) × 4
•
4230 (4 + 2) × 3 + 0 = . 6815 6 + 8 + 15
•
3160 (3 + 1) × 6 + 0 = . 8295 8 + (2 + 9) × 5
•
4280 4 × (2 + 8) + 0 = . 5671 5 + 6 × (7 + 1)
•
4360 4 + 36 + 0 = . 8175 (8 + 1 × 7) × 5
•
5130 5 + 130 = . 9728 (9 + 7) × 2 × 8
•
5370 53 + 7 + 0 = . 6981 6+9×8×1
•
5610 5×6×1+0 = . 7293 7 + 29 + 3
•
5620 5 × (6 + 2) + 0 = . 8149 8 + 1 + 49
•
5×6×4+0 5640 = . 7238 7 × 2 × (3 + 8)
•
•
1580 1×5×8+0 = . 3792 3 × (7 + 9) × 2 1590 1+5+9+0 = . 3286 3 + 2 × (8 + 6) 1 × 8 + 20 1820 = . 6435 64 + 35 1840 1×8×4+0 = . 3795 3×7+9×5 1860 1+8+6+0 = . 3472 (3 + 4 + 7) × 2
•
1870 18 + 7 + 0 = . 5236 52 + 3 × 6 1×8×9+0 1890 = . 3465 3 × 4 × (6 + 5)
1890 18 × 9 + 0 • = . 3675 (3 + 6) × 7 × 5 1890 18 + 90 • = . 4375 (43 + 7) × 5
•
•
1980 (1 + 9) × 8 + 0 = . 3267 3 × (2 + 6 × 7) 1980 (1 + 9) × 8 + 0 = . 6237 6×2×3×7
•
2970 29 + 7 + 0 = . 3465 3×4+6×5
3190 3 + 19 + 0 = . • 7685 7+6+8×5
1980 1×9×8+0 = . 6435 6 × (4 + 35)
•
2310 2 + 3 + 10 • = . 7854 7+8×5+4
•
•
2690 2 × (6 + 9) + 0 = . 4573 4 × (5 + 7) + 3
2890 2 × (8 + 9) + 0 • = . 6375 63 + 7 + 5
• •
•
•
3240 3 × 24 + 0 = . 7695 76 + 95 3270 3 + 27 + 0 = . 9156 (9 + 1 × 5) × 6
•
1520 15 × 2 + 0 = . 3496 (3 + 4) × 9 + 6
•
2340 2 + 34 + 0 = . 7865 7 × 8 + 65
•
3420 34 + 2 + 0 = . 6175 (6 + 1 × 7) × 5
•
1540 15 × 4 + 0 = . 2387 2 × 3 + 87
•
2350 (2 + 3) × 5 + 0 = . 7614 76 + 1 + 4
•
3 + 42 + 0 3420 = . 9576 (9 + 5 + 7) × 6
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
5840 5 × 8 + 40 = . 9271 9×2×7+1
•
140 1×4+0 = . 37625 (37 + 6) × 25
•
510 5×1+0 = . 29376 2 × 9 × (3 + 7 + 6)
•
630 6×3+0 = . 12985 1 + (2 + 9 × 8) × 5
•
5910 59 + 1 + 0 = . 7486 7×4+8×6
•
140 1+4+0 = . 79632 79 × 6 × 3 × 2
•
510 5×1+0 = . 29784 (2 + 9 × 7 + 8) × 4
•
630 6×3+0 = . 14875 (1 + (4 + 8) × 7) × 5
•
5×9×4+0 5940 = . 8613 (86 + 1) × 3
•
170 1+7+0 = . 63495 6 × (3 + 495)
510 5×1+0 • = . 39678 3+9×6×7+8
•
6+9+0 690 = . 13248 1 × 3 × 2 × 48
•
6120 6 × 12 + 0 = . 7395 73 + 9 + 5
•
6×9+0 690 = . 14375 (1 + 4) × 3 × 75
•
6280 6 × (2 + 8) + 0 = . 9734 9+7×3×4
•
690 6+9+0 = . 15732 1 × 57 × 3 × 2
•
6+9+0 690 = . 23184 2 × 3 × 1 × 84
•
7+1+0 710 = . 48635 4 + 8 × 63 + 5
6×3+2+0 6320 = . • 7584 7+5+8+4
210 2×1+0 • = . 37485 3 × 7 × (4 + 8 + 5)
•
•
•
•
•
•
•
230 2×3+0 = . 94875 9 × (48 + 7) × 5
•
5×2+0 520 = . 61984 6 × 198 + 4
•
730 7+3+0 = . 14892 14 × 8 + 92
•
260 2+6+0 = . 43875 (4 + 38 × 7) × 5
•
520 5×2+0 = . 78936 7×8×9×3+6
•
810 8 + 10 = . 43965 4 × 3 + 965
•
310 3+1+0 = . 69285 6 × (9 + 28 × 5)
•
530 5×3+0 = . 18974 1 + 8 × (9 × 7 + 4)
•
8+2+0 820 = . 13694 1 + (3 × 6 × 9 + 4)
•
3+2+0 320 = . 19648 19 + 6 × 48
•
530 5 + 30 = . 79182 7 × 9 × (1 + 82)
•
820 8+2+0 = . 35916 ((3 + 5) × 9 + 1) × 6
•
8+2+0 820 = . 37146 3 + (71 + 4) × 6
•
870 8+7+0 = . 19546 1 + (9 + 5) × 4 × 6
•
910 9+1+0 = . 28574 2 + 8 × (5 × 7 + 4)
•
•
910 9+1+0 = . 32487 (3 + (2 + 4) × 8) × 7
•
940 9×4+0 = . 21385 21 × 3 × (8 + 5)
•
960 9+6+0 = . 18432 1 × 8 × (4 + 32)
540 5 + 40 = . 89712 89 × 7 × 12
•
1260 1×2×6+0 = . 37485 3 × 7 × (4 + 8 + 5)
6840 6 × (8 + 4) + 0 = . 9215 92 + 1 × 5 6930 6 + 9 + 30 = . 7854 7+8×5+4 (7 + 1) × 2 + 0 7120 = . 9345 9+3+4+5
7260 72 + 60 = . 9185 9 × 18 + 5 7320 7 + 3 + 20 = . 9516 9+5×1×6
540 5 + 40 • = . 12768 (127 + 6) × 8
370 3+7+0 • = . 14985 (1 + 49) × 8 + 5
540 5 + 40 • = . 18792 1 × 87 × 9 × 2
370 3+7+0 • = . 28194 2 + 8 × (1 + 94)
•
7860 7 + 8 + 60 = . 9432 9 × (4 + 3 × 2)
•
•
7910 7 × (9 + 1) + 0 = . 8362 8 × (3 + 6) + 2
•
•
8720 8 + 72 + 0 = . 9156 (9 + 1 × 5) × 6
•
•
•
510 5×1+0 = . 76398 7 × (6 + 3 + 98)
340 3×4+0 • = . 98175 (98 + 1) × 7 × 5
7830 78 + 30 = . 9425 (9 + 4) × 2 × 5
•
510 5×1+0 = . 67932 6 × (79 + 32)
•
6840 6×8×4+0 = . 7125 (7 + 1) × 25
7230 72 + 3 + 0 = . 8194 (8 + 1) × 9 + 4
•
230 2+3+0 = . 17986 17 × (9 + 8 + 6)
•
6340 6×3×4+0 • = . 7925 (7 + 9 + 2) × 5
•
210 2×1+0 = . 64785 6 + 47 × (8 + 5)
510 5 + 10 • = . 43792 (4 + 3 + 7) × 92
380 3×8+0 = . 29165 2 × (916 + 5)
•
•
540 5×4+0 = . 18927 1 + (8 + 92) × 7 540 5×4+0 = . 26973 26 + 973
410 4+1+0 = . 39852 3 × 9 × (8 + 5 × 2)
•
430 4×3+0 = . 26875 2 × (68 + 7) × 5
•
•
430 4×3+0 = . 69875 6 × (9 + 8 × 7) × 5
•
620 6+2+0 = . 13485 134 + 8 × 5
•
1320 (1 + 3) × 2 + 0 = . 48675 (4 + 8 × 6 + 7) × 5
130 1+3+0 = . 24895 2 + 4 + 8 × 95
•
460 4+6+0 = . 15732 1 × 57 × 3 × 2
•
620 6+2+0 = . 34875 (34 + 8 × 7) × 5
•
1370 1×3+7+0 = . 28496 28 × 4 + 96
130 1+3+0 = . 26975 2 + 69 × (7 + 5)
•
460 4+6+0 = . 23598 23 + 5 × 98
•
620 6×2+0 = . 81375 (8 + 13) × 75
•
1 + 4 + 20 1420 = . 57936 5 × (7 + 9 × 3) × 6
540 5×4+0 = . 32967 3 + 29 × 6 × 7
Inder J. Taneja
•
1430 1 + 43 + 0 = . 26975 2 + 69 × (7 + 5)
1450 (1 + 4) × 5 + 0 • = . 39672 3 + 9 + 672
•
•
1460 14 + 6 + 0 = . 89352 8 × 9 × (3 × 5 + 2) 1460 1×4+6+0 = . 97382 9 × 73 + 8 + 2
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
1640 1 × 64 + 0 • = . 78925 7 × 8 × (9 + 2) × 5
•
•
1470 14 + 70 = . 26985 2 × (6 + 9 × 85)
•
•
1470 (1 + 4) × 7 + 0 = . 29568 (2 + 9) × (56 + 8)
•
•
•
•
•
(2 + 3) × 4 + 0 2340 = . • 75816 (7 + 5) × (8 + 1) × 6
•
•
2340 (2 + 3) × 4 + 0 = . 85176 8 × (5 × 17 + 6) 2×3+9+0 2390 = . 48756 4 × (8 + 7) × 5 + 6
•
2790 27 + 9 + 0 = . 13485 134 + 8 × 5
•
2790 2+7+9+0 = . 43865 43 + 8 × 6 × 5
•
2810 2 + 8 + 10 = . 37654 (37 + 6 × 5) × 4
•
2840 28 × 4 + 0 = . 17395 1 × 7 × (3 + 95)
•
2840 2 + 8 + 40 = . 57936 5 × (7 + 9 × 3) × 6
•
2850 (2 + 8) × 5 + 0 = . 13794 1 + 3 × 79 + 4
•
1820 1 × 8 + 20 = . 64935 64 + 935
•
2430 24 + 30 = . 16875 1 × (68 + 7) × 5
•
2850 (2 + 8) × 5 + 0 = . 74613 7 × (4 + 61 × 3)
1 + (4 + (7 + 0)) 1470 = . 69825 6 × (9 + 8 + 2) × 5
•
1870 1 + 87 + 0 = . 63495 6 × (3 + 495)
•
2430 2 + 43 + 0 = . 71658 7 + 165 × 8
•
2870 28 + 7 + 0 = . 13694 1+3×6×9+4
1480 1×4×8+0 = . 37925 (3 + 79) × 2 × 5
•
1890 1+8+9+0 = . 42735 42 + 73 × 5
24 + 6 + 0 2460 = . 17958 179 + 5 × 8
•
•
28 + 7 + 0 2870 = . 35916 ((3 + 5) × 9 + 1) × 6
1520 1×5×2+0 = . 36784 3×6+7×8×4
(1 + 9) × 2 + 0 1920 = . • 75648 756 + 4 × 8
•
2890 (2 + 8) × 9 + 0 = . 16473 (164 + 7) × 3
•
2940 2+9+4+0 = . 15876 1 × (5 × (8 + 7) + 6)
•
2 × 96 + 0 2960 = . 13875 (1 + 3 + 8) × 75
2460 2+4+6+0 = . 89175 (8 × (9 + 1) + 7) × 5
•
3170 3 + 17 + 0 = . 58962 5×8×9+6×2
1×5×3+0 1530 = . 29784 (2 + 9 × 7 + 8) × 4
•
(1 + 9) × 3 + 0 1930 = . 25476 2 × (5 + 4 × 7) × 6
2410 24 + 1 + 0 = . 59768 (5 + 97) × 6 + 8
2 × (4 + 6) + 0 2460 = . • 31857 ((3 + 1) × 8 + 5) × 7
•
2460 2 × (4 + 6) + 0 = . 58179 58 × (1 + 7) + 9
1950 (1 + 9) × 5 + 0 = . 48672 48 × (6 + 7) × 2
•
•
(1 + 9) × 5 + 0 1950 = . 78624 7 × 8 × 6 × (2 + 4)
•
2470 2×4+7+0 = . 95836 (9 + 5 + 83) × 6
•
3170 3+1×7+0 = . 92564 9 × (2 + 5 × 6) + 4
1540 1×5×4+0 = . 76923 76 + 923
•
2140 21 × 4 + 0 = . 37985 3 × (7 + 98 × 5)
•
2510 2×5×1+0 = . 36897 3 + 6 × (8 + 9 + 7)
•
3190 3 × (1 + 9) + 0 = . 75284 (7 + 52) × (8 + 4)
1540 1 + 54 + 0 = . 79632 79 × 6 × 3 × 2
•
2 × 16 + 0 2160 = . 43875 (43 + 87) × 5
•
2510 2×5×1+0 = . 46937 4 × (6 + 9) × 3 + 7
•
3210 3+2×1+0 = . 87954 8 × (7 + 9) + 5 + 4
•
3+2×1+0 3210 = . 97584 9 × 7 + 5 + 84
•
3240 3 + 2 + 40 = . 15768 1+5×7×6+8
•
3 × 24 + 0 3240 = . 16875 1 × (68 + 7) × 5
•
1540 1+5+4+0 = . 37268 3 × (72 + 6) + 8
•
•
1580 1×5×8+0 • = . 79632 (7 + 9) × 63 × 2 1590 1+5+9+0 = . 37842 3 × 7 + 8 × 42
2170 2×1×7+0 • = . 43865 43 + 8 × 6 × 5
•
1590 1 + 59 + 0 = . 47382 47 × 38 + 2
•
•
1630 1+6+3+0 = . 54279 (5 × (4 + 2) + 7) × 9
•
16 + 4 + 0 1640 = . 39852 3 × 9 × (8 + 5 × 2)
•
1720 1+7+2+0 = . 59684 5 + 9 × (6 + 8 × 4)
2310 2 + 3 + 10 = . 78694 7 + (8 + 6) × 9 × 4
•
•
•
1720 1+7+2+0 = . 35948 3 × 59 + 4 × 8
•
1730 1×7+3+0 = . 85462 8 × 54 + 62
1540 15 × 4 + 0 = . 26873 2 × 6 × 87 + 3
•
1640 1+6×4+0 = . 53792 5 × (3 + 79) × 2
2190 2 × (1 + 9) + 0 = . 73584 7 × (3 + 5) × (8 + 4)
2510 2 × 5 + 10 • = . 98643 9 × 86 + 4 × 3
•
2 + 53 + 0 2530 = . 17986 17 × (9 + 8 + 6)
2310 2+3×1+0 = . 47586 (4 + 7) × 5 + 8 × 6
•
2580 2+5+8+0 = . 19436 1 + 94 + 3 × 6
•
3240 3 + 2 + 40 = . 17856 1+7+8×5×6
•
2310 2+3×1+0 = . 54978 5 + 4 × 9 + 78
•
2640 2×6+4+0 = . 98175 (9 + 8 × 1) × 7 × 5
•
3240 3 + 2 + 40 = . 18576 1 × (8 + 5 × 7) × 6
•
2310 2+3×1+0 = . 58674 5 + 8 × 6 + 74
•
2730 27 + 3 + 0 = . 18564 ((1 + 8) × 5 + 6) × 4
•
(3 + 2) × 4 + 0 3240 = . 75168 (75 + 1) × 6 + 8
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
3270 3+2+7+0 = . 16895 1×6×8+9+5
•
3620 3×6+2+0 = . 51947 (5 + 1 × 9 × 4) × 7
•
4170 (4 + 1) × 7 + 0 = . 28356 28 + 35 × 6
•
4710 4+7+1+0 = . 36895 3 + 6 + (8 + 9) × 5
•
3270 3+2+7+0 = . 41965 (4 + 1 + 9) × (6 + 5)
•
3640 36 + 4 + 0 = . 72891 (7 + 2) × 89 × 1
•
4170 4+1+7+0 = . 93825 9 + 3 × (82 + 5)
•
4780 47 + 8 + 0 = . 15296 1 + 5 × (29 + 6)
•
4860 4 + 8 + 60 = . 37125 (3 × 7 + 1) × 25
•
4920 4 × 9 + 20 = . 17835 1 × 7 × (8 × 3 + 5)
•
4960 4 × (9 + 6) + 0 = . 15872 15 × 8 + 72
•
5160 5 × 16 + 0 = . 24897 2 + 4 × (89 + 7)
3×2×8+0 3280 = . • 19475 (1 + 9 + 47) × 5
3640 36 + 4 + 0 • = . 91728 9×1×7×2×8
4180 4 × 18 + 0 • = . 67925 (6 + 7) × 9 × 2 × 5
3290 (3 + 2) × 9 + 0 • = . 46718 (4 + 67) × (1 + 8)
3690 3+6+9+0 • = . 18245 1 + 8 × (2 + 4 + 5)
4190 4 × (1 + 9) + 0 • = . 76258 7 × (6 + 2 + 5) × 8
•
•
3720 (3 + 7) × 2 + 0 = . 45198 45 + 198
•
4260 4+2+6+0 = . 17395 1 × 7 + 3 × (9 + 5)
•
5170 5 + 1 × 70 = . 49632 4 × 9 × (6 × 3 + 2)
3 × (4 + 6) + 0 3460 = . 21798 2 + 179 + 8
•
3720 (3 + 7) × 2 + 0 = . 81654 (81 + 6) × 5 + 4
•
4280 4 × (2 + 8) + 0 = . 61953 6 × (1 + 95) + 3
•
5 + 180 5180 = . 79632 79 × 6 × 3 × 2
3460 34 + 6 + 0 = . 87192 8×7×1×9×2
•
3720 (3 + 7) × 2 + 0 = . 95418 95 + 418
•
4 × (3 + 2) + 0 4320 = . 15768 1 × 5 × (7 + 6) + 8
•
5260 5×2×6+0 = . 39187 391 + 8 × 7
•
5280 5 + 2 × 80 = . 14976 1 × 4 × 9 × (7 + 6)
•
5290 5 × (2 + 90) = . 73186 (73 + 1) × 86
•
5310 5×3×1+0 = . 94872 94 + 87 × 2
•
5340 5×3×4+0 = . 71289 (7 + 1 × 2) × 89
•
5340 5×3×4+0 = . 72891 728 + 91
•
5340 5×3×4+0 = . 89712 8×9×7×1×2
•
5360 5 × (3 + 6) + 0 = . 81472 (8 + 1) × (4 + 72)
•
5410 5×4×1+0 = . 28673 2 × (8 + 6 × 7 + 3)
•
5420 5×4×2+0 = . 19783 1 × 9 × 7 + 83
•
5430 5 × (4 + 30) = . 79821 7 × (9 + 8) × 21
•
5460 5 + 4 × 60 = . 13728 (1 + 3 × 7) × 28
•
5460 54 + 6 + 0 = . 72891 (7 + 2) × 89 × 1
•
5×4×9+0 5490 = . 16287 1 × 6 × (2 + 87)
•
•
3410 3 + 41 + 0 = . 69285 6 × (9 + 28 × 5)
•
•
3480 3+4+8+0 • = . 65192 (6 × 5 + 1) × 9 + 2
•
3480 (3 + 4) × 8 + 0 = . 76125 7 × (6 + 1) × 25
3780 3 × (7 + 8) + 0 • = . 14952 14 × 9 + 52 3780 3 × (7 + 8) + 0 • = . 15624 (1 + 5 × 6) × (2 + 4)
4520 4×5×2+0 = . 16837 16 × 8 + 3 × 7 4 × 5 + 60 4560 = . 32718 3 × 2 + 71 × 8
•
3780 3 × (7 + 8) + 0 = . 64512 64 × (5 + 1) × 2
•
4620 4 × 6 + 20 = . 91875 (9 + 1) × 87 + 5
3520 3 + 52 + 0 = . 19648 19 + 6 × 48
•
3790 3×7+9+0 = . 48512 4 × 8 × (5 + 1) × 2
•
4620 4 × (6 + 2) + 0 = . 98175 (9 + 8) × (1 + 7) × 5
3540 3×5×4+0 = . 71862 7 × (1 + 86) × 2
•
38 + 2 + 0 3820 = . 17954 1+7+9×5×4
•
4630 4 + 6 + 30 = . 15279 1+5+2×7×9
•
3510 3×5×1+0 = . 76284 7 × 6 + 284
•
•
3570 3 × 5 + 70 • = . 14826 1 + 4 × (82 + 6)
3860 3×8+6+0 • = . 72954 (7 + 2) × (9 + 54)
3570 3×5×7+0 = . 82416 8 + 2416
•
3610 3+6+1+0 = . 42598 4 + 2 + (5 + 9) × 8
•
•
3610 3+6+1+0 = . 54872 (5 × 4 + 8 × 7) × 2
•
3610 3+6+1+0 = . 94582 9 × 4 × 5 + 82
•
•
•
•
•
4390 4 × (3 + 9) + 0 • = . 76825 (76 + 8) × 2 × 5
3780 3 + 7 + 80 = . 52416 52 × 4 × 1 × 6
3×5×1+0 3510 = . 48672 4 × (8 + 6 × 7 + 2)
•
4230 42 + 3 + 0 = . 61758 617 + 5 × 8
3 × (6 + 9) + 0 3690 = . 25748 2 + (5 × 7 + 4) × 8
3290 3+2+9+0 = . 86715 8 + 6 + 71 × 5
•
3870 3 + 87 + 0 = . 16942 16 + 9 × 42
(4 + 6) × 3 + 0 4630 = . • 85192 8 × (51 + 9 × 2)
•
4680 (4 + 6) × 8 + 0 = . 13572 1 + 3 × (5 + 72)
3960 3 × (9 + 6) + 0 = . 14872 1 + (4 + 8) × 7 × 2
•
•
4150 (4 + 1) × 5 + 0 = . 37682 3 × (7 + 68) + 2
•
4680 (4 + 6) × 8 + 0 = . 73125 (7 + 3) × 125
•
4 + 1 × 60 4160 = . 93275 (9 + 32) × 7 × 5
•
4 + 68 + 0 4680 = . 93275 (9 + 32) × 7 × 5
4680 4 + 68 + 0 = . 13975 ((1 + 3) × 9 + 7) × 5
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
5620 5 × (6 + 2) + 0 = . 14893 1 + 4 + 8 + 93
•
6390 6+3+9+0 = . 47215 4 × 7 + 21 × 5
•
7380 7 + 3 + 80 = . 12546 (1 + 2) × (5 + 46)
•
5720 5 × 7 + 20 = . 93184 (9 × 3 + 1) × 8 × 4
•
6420 6 × 42 + 0 = . 37985 3 × (7 + 98 × 5)
•
7380 7 + 38 + 0 = . 12956 1 + 2 × (9 + 5 × 6)
5730 57 + 3 + 0 • = . 19482 1 × (94 + 8) × 2
•
5×7×4+0 5740 = . 23698 2 + (3 + 69) × 8
6430 6 + 4 + 30 • = . 29578 2 × (9 + 5 + 78)
•
5780 5+7+8+0 = . 13294 1 + 3 × (2 + 9 + 4)
•
•
5+7+8+0 5780 = . 32946 3 × 2 × (9 + 4 + 6)
•
•
•
•
•
(7 + 9) × 6 + 0 7960 = . 34825 (34 + 8) × 2 × 5
•
7980 7 + 98 + 0 = . 13452 (1 + 34) × 5 + 2
•
7980 7 + 98 + 0 = . 46512 (46 + 5) × 12
•
8190 8+1+9+0 = . 63245 6 + 32 × 4 + 5
7 × 4 + 10 7410 = . 23985 2 × 3 + 9 × (8 + 5)
6540 6+5+4+0 = . 83712 8 × (3 + 7 × (1 + 2))
•
7480 (7 + 4) × 8 + 0 = . 16235 1 × 62 × 3 + 5
•
8240 8×2+4+0 = . 13596 13 + 5 + 9 + 6
5810 5 × (8 + 10) = . 34279 (3 + 4 × 2 × 7) × 9
•
6×5×7+0 6570 = . 14892 14 × (8 + 9) × 2
•
7480 7 + 48 + 0 = . 16592 1 + (6 + 5) × (9 + 2)
•
8460 84 + 6 + 0 = . 15792 1 × (5 + 79) × 2
5840 5 × (8 + 4) + 0 = . 36792 367 + 9 + 2
•
6590 6+5+9+0 = . 78421 7 × (8 × 4 + 2 × 1)
(7 + 4) × 8 + 0 7480 = . • 19635 196 + 35
•
8460 8 + 4 + 60 = . 29375 (29 + 3 × 7) × 5
6810 6×8×1+0 = . 39725 39 × 7 + 2 + 5
7+5+3+0 7530 = . • 29618 2+9+6×1×8
•
8 + 47 + 0 8470 = . 12936 1 + 2 + 9 × (3 + 6)
•
8470 8 × (4 + 7) + 0 = . 19635 196 + 3 + 5
•
8520 8 × (5 + 20) = . 14697 (1 + 4) × (6 + 9 × 7)
7560 7 × (5 + 60) = . 13824 13 × 8 × 2 × 4
•
8 + 6 + 10 8610 = . 93275 (9 × (3 + 2) + 7) × 5
•
8670 8 + 67 + 0 = . 13294 1 + 3 × (2 + 9 × 4)
•
8670 8+6×7+0 = . 25143 2 × 51 + 43
•
(8 + 7) × 6 + 0 8760 = . 54312 (5 + 4) × 31 × 2
•
(9 + 1) × 5 + 0 9150 = . 38247 (3 + 8 × 2) × (4 + 7)
•
9170 (9 + 1) × 7 + 0 = . 38645 (3 + (8 + 6) × 4) × 5
•
9170 (9 + 1) × 7 + 0 = . 45326 4 × 5 + 326
•
9170 (9 + 1) × 7 + 0 = . 58426 5 × 84 + 26
•
9180 9×1×8+0 = . 62475 (6 + 2 × 4) × 7 × 5
•
9 + 1 + 80 9180 = . 74256 (7 + 4 + 2) × 56
5840 5×8×4+0 = . 63291 6 × (32 × 9 + 1)
•
•
•
5970 5 × (9 + 7) + 0 = . 14328 1×4×3×2×8
•
6820 6 × (8 + 2) + 0 = . 19437 1 × 9 × (4 × 3 + 7)
•
7460 7 × 4 + 60 = . 83925 (8 + 3) × 9 × 2 × 5
7540 7 × 5 + 40 = . 36192 3 × 6 × (1 + 9) × 2
6890 6 × 8 + 90 = . 72345 7 × 23 × (4 + 5)
•
•
6930 6 + 9 + 30 = . 17248 (1 + 7 + 2 + 4) × 8
•
7×5×6+0 7560 = . 23184 2 × (318 + 4)
6170 61 + 7 + 0 = . 83295 83 × (2 + 9) + 5
•
6930 6 × (9 + 3) + 0 = . 48125 (4 + 8 × 12) × 5
•
7590 75 + 90 = . 13248 1 × 3 × 2 × 48
•
6180 6 + (1 + 8) + 0 = . 45732 4+5×7×3+2
•
7140 7×1×4+0 = . 29835 29 + 83 + 5
•
75 + 90 7590 = . 14628 (1 + 4) × 62 + 8
•
6 + 210 6210 = . 94875 (9 × 4 + 8) × 75 6280 6+2+8+0 = . 71435 7 + (1 + 4) × 35
7×1×4+0 7140 • = . 32895 3 × 28 + 9 × 5
•
6 × 2 + 90 6290 = . 13875 1 × 3 × (8 + 7) × 5
•
•
6380 6 + 38 + 0 = . 24795 2 × (4 + 79) + 5
•
6 + 39 + 0 6390 = . 21584 (2 × 15 + 8) × 4
•
7910 7 × (9 + 1) + 0 = . 35482 (35 + 4) × 8 + 2
•
•
•
•
•
5940 5 + 940 = . 13728 1 × 3 × 728
•
7840 (7 + 8) × 4 + 0 = . 12936 1 × (2 + 9) × (3 + 6)
6530 6×5×3+0 = . 41792 4 × (1 + 7) × 9 × 2
5 × (8 + 4) + 0 5840 = . • 39712 (3 × 9 + 7) × 12
•
6480 (6 + 4) × 8 + 0 = . 17253 (1 + 7 × 2 × 5) × 3
7380 7 + 38 + 0 • = . 14596 1 + 4 + (5 + 9) × 6
•
7×2×1+0 7210 = . 83945 8 + (3 × 9 + 4) × 5
7590 75 + 90 • = . 23184 2 × 3 × 1 × 84
•
7590 75 + 90 = . 61824 (6 + 1) × 8 × 24 7680 7×6+8+0 = . 14592 1+4+5×9×2
7240 7+2×4+0 = . 31856 (3 + 1 + 8) × 5 + 6
•
•
7280 7×2×8+0 = . 64935 64 + 935
•
7810 7 + 8 + 10 = . 35926 3 + (5 + 9) × (2 + 6)
•
7320 (7 + 3) × 2 + 0 = . 65148 6 × 5 + 148
•
7+8+3+0 7830 = . 29145 2 + (9 + 1 × 4) × 5
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
9210 9 + 21 + 0 = . 46357 4 + ((6 + 3 × 5) × 7)
•
410 4×1+0 = . 379865 3 + 7 × (9 + 8 × 65)
•
9320 (9 + 3) × 2 + 0 = . 61745 6 + 17 × (4 + 5)
•
870 8+7+0 420 4×2+0 = . = . • 129456 12 × (9 × 4 × 5 + 6) 196875 (1 + 9) × (68 + 7) × 5
9420 9+4+2+0 • = . 35168 3+5+1×6×8
•
9460 9 + 46 + 0 = . 37152 (3 + 7 × 15) × 2
460 4×6+0 • = . 129375 1 × 2 × 9 × 375
•
15390 15 + 3 × 90 = . 26784 2 × (6 + 7 × 8) × 4
•
15790 15 × (7 + 9) + 0 = . 26843 (26 + 8) × 4 × 3
•
15 × 8 × 4 + 0 15840 = . 32967 32 + 967
•
15840 1×5×8×4+0 = . 76923 7 × (6 × 9 × 2 + 3)
•
(1 + 6 × 3) × 4 + 0 16340 = . 97825 (9 + 7 × 8) × (2 + 5)
910 9+1+0 = . 438256 43 × (82 + 5 × 6)
•
16380 (1 + 6) × 3 × 8 + 0 = . 27495 2 + (7 + 49) × 5
870 8+7+0 • = . 219356 2 × (1 + 9 × 35 × 6)
•
9+1+0 910 = . 267358 26 × (7 × 3 × 5 + 8)
•
510 5 + 10 = . 389674 3 + (8 + 9) × 674
•
910 9+1+0 = . 657384 (65 + 7 × 3) × 84
•
16920 (1 + 69) × 2 + 0 = . 83754 (8 + 3) × 7 × (5 + 4)
9560 9+5+6+0 = . 24378 2 + 4 + 3 × (7 + 8)
•
5 × (2 + 0) 520 = . 481936 4 + 8 × 193 × 6
•
12360 1 × 2 × 36 + 0 = . 84975 (8 + (4 + 9) × 7) × 5
•
17 + 28 + 0 17280 = . 49536 (4 × 9 + 5) × 3 + 6
9×5×7+0 9570 = . 61248 (61 + 2) × 4 × 8
•
530 5×3+0 = . 279416 2 × (7 × 94 + 1) × 6
•
1 × 28 × 6 + 0 12860 = . 73945 7 × 3 + 945
•
17380 1 × (7 + 3) × 8 + 0 = . 29546 2 × 9 × 5 + 46
•
178 + 2 + 0 17820 = . 43659 (4 + (3 + 6) × 5) × 9
•
17820 17 + 8 + 20 = . 93456 9 + 3 + 4 × 56
•
17860 1 + 7 + 8 + 60 = . 39245 3 × 9 × (2 + 4) + 5
13470 1+3×4×7+0 = . 59268 (59 + 2) × 6 + 8
•
1 + 7 × 9 + 40 17940 = . 28635 2 × (8 × 6 + 35)
•
9540 9×5×4+0 = . 28673 2 + 8 × 67 + 3
•
•
9650 (9 + 6) × 5 + 0 • = . 12738 1 × (2 + 7) × (3 + 8)
•
9680 96 + 80 = . 13475 1 × (3 + 4) × 7 × 5
•
•
9680 96 + 80 = . 47135 4 × 71 × 3 + 5
•
•
9780 97 + 8 + 0 = . 36512 (3 × 65 + 1) × 2
•
•
540 5 + 40 = . 168792 16 × 879 + 2 540 5×4+0 = . 813726 81 × 372 + 6
•
•
(1 + 2) × 96 + 0 12960 = . 38475 3 + 847 + 5 12960 1 × 2 × 96 + 0 = . 43875 (43 + 87) × 5
540 5×4+0 = . 862137 (862 + 1) × 37
•
•
560 5 × (6 + 0) = . 972384 9 × (723 × 8 + 4)
•
13620 1+3+6×2+0 = . 98745 9 + 87 + 4 × 5
•
17940 1 + 7 × 9 + 40 = . 63825 ((6 + 3) × 8 + 2) × 5
9810 9+8+1+0 = . 46325 4 + 6 + 3 × 25
•
620 6×2+0 = . 194835 1 × 9 × (4 + 83 × 5)
•
13960 (1 + 39) × 6 + 0 = . 42578 4 × (25 × 7 + 8)
•
18270 1 × 8 + 27 + 0 = . 35496 3 + 5 + 4 × (9 + 6)
140 1+4+0 = . 683592 68 × 359 + 2
•
690 6+9+0 = . 185472 (1 + 8 + 5) × 4 × 72
•
14320 1×4×3+2+0 = . 75896 7 × (5 + 8) + 9 + 6
•
1 + 8 × 3 + 60 18360 = . 45792 4 × (5 × 7 + 9 × 2)
•
18360 18 × 3 + 6 + 0 = . 79254 7 × (9 + (2 + 5) × 4)
•
18360 1×8×3+6+0 = . 95472 (9 + 5) × (4 + 7) + 2
•
18640 1 × 8 × (6 + 4) + 0 = . 25397 25 + (3 + 9) × 7
•
18720 1 × (8 + 72) + 0 = . 59436 59 × 4 + 3 × 6
•
19320 1 × (9 + 3) × 2 + 0 = . 74865 74 + 8 + 6 + 5
•
19350 1+9+3×5+0 = . 87462 87 + 4 × 6 + 2
170 1+7+0 • = . 469285 4 + 6 × 92 × 8 × 5
690 6+9+0 • = . 217534 21 × 75 × 3 + 4
210 2×1+0 7+3+0 • = . 730 398475 (3 + 9 × (8 + 4) × 7) × 5• 142569 = (1 + 42 × 5 + 6) × 9 .
•
•
•
•
•
810 8×1+0 = . 374625 37 × (4 + 6) × 2 × 5
4×8+0 480 = . 219375 (2 + 193) × 75
95 + 40 9540 = . 21836 21 + 8 × 36
•
460 4 + 60 = . 139725 (1 + 3) × 972 × 5
•
(1 + 4) × 35 + 0 14350 • = . 69782 69 + 782 1+4+5×8+0 14580 • = . 23976 2+3+9×7+6
14630 1 × 46 + 30 230 2+3+0 730 7+3+0 • = . = . 92785 92 + 78 × 5 158746 1 + (5 × (8 + 7) × 46) • 149285 = (1 + (49 + 2) × 8) × 5 .
•
14760 1 × 4 + 76 + 0 230 2+3+0 7+3+0 • = . = . • 730 = . 39852 (3 + 9) × (8 + 5 × 2) 158976 (1 + 5) × (89 + 7) × 6 215496 2 × ((1 + 5 × 49) × 6)
•
230 2+3+0 = . 894516 (8 × 9 × 45 + 1) × 6
•
790 7+9+0 = . 361425 3 × 61 × 4 × 2 × 5
•
1×5×2×8+0 15280 = . 63794 6 + (3 + 79) × 4
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
19380 19 × 3 × 8 + 0 = . 62475 6 × (2 + 47) × 5
•
26890 2+6+8×9+0 = . 45713 45 + 7 × 13
•
35640 3×5×6×4+0 = . 87219 872 + 1 × 9
•
41580 415 + 80 = . 79632 79 × (6 + 3 × 2)
•
19530 (1 + 9) × (5 + 30) = . 26784 (2 + 6 + 7) × 8 × 4
•
26910 26 × (9 + 1) + 0 = . 87354 8×7×3×5+4
•
35640 35 + 640 = . 91872 (9 + 1) × 87 × 2
•
41860 4 × 1 × 8 + 60 = . 93275 (9 × 3 + 2 × 7) × 5
3 × (6 + 2) + 10 36210 = . • 94785 9 + 4 × (7 + 8 + 5)
•
(4 + 2 × 3) × 5 + 0 42350 = . 79618 79 + 6 + 1 + 8
•
42360 4 × (2 + 3 × 6) + 0 = . 75189 7 + 5 × (18 + 9)
•
43 × 5 + 60 43560 = . 91872 (9 + 1) × (8 × 7 + 2)
•
43710 4 × 3 × 7 + 10 = . 69285 6 + (9 + 2) × (8 + 5)
•
45360 4 × 5 × 36 + 0 = . 98721 9 × 87 × 2 + 1
•
45960 45 + 9 + 6 + 0 = . 71238 71 + 2 × (3 + 8)
•
46170 46 + 1 + 7 + 0 = . 98325 (9 + 8 + 3 × 2) × 5
•
(4 + 6 × 1) × 9 + 0 46190 = . 78523 (7 + 8) × 5 × 2 + 3
•
46230 46 × (2 + 3) + 0 = . 71958 7 × (1 + 9) × 5 + 8
•
46320 (4 + 6) × 32 + 0 = . 97851 9 × (7 + 8) × 5 + 1
•
4 × 69 × 2 + 0 46920 = . 73185 7 × 3 × (1 + 8 × 5)
•
47310 4 × (73 + 10) = . 86925 (8 × (6 + 9) + 2) × 5
•
47960 4 + 7 + 9 + 60 = . 81532 8 × (1 + (5 + 3) × 2)
19720 (1 + 9 × 7) × 2 + 0 • = . 46835 4 × (68 + 3 + 5)
•
(1 + 9) × 75 + 0 19750 = . 34286 3 × (428 + 6)
2 × (6 + 9) + 30 26930 = . • 45781 45 + 7 × 8 + 1 27390 2 × (7 + 3) + 90 • = . 45816 4 × (5 × 8 × 1 + 6)
•
•
2 + 7 + 81 + 0 27810 = . 54693 (5 × (4 + 6) + 9) × 3
•
36720 36 + 7 × 20 = . 84915 8 × 49 + 15
21780 2 + 1 × 78 + 0 = . 93654 9 × 36 + 5 × 4
•
2 + 79 + 3 + 0 27930 = . 61845 6 + (1 + 8) × 4 × 5
•
37260 3 × 72 + 60 = . 49815 4 + (9 × 8 + 1) × 5
21870 2 + 1 + 87 + 0 = . 35964 (3 × 5 + 9) × 6 + 4
•
28350 (2 + 8) × 35 + 0 = . 49167 4 + 9 × 1 × 67
•
3 + 7 + 2 + 60 37260 = . 94185 (9 + 4 + 1) × (8 + 5)
•
•
21670 2 + 16 + 7 + 0 = . 95348 9+5+3×4×8
•
•
21870 2 × 1 × (8 + 7) + 0 • = . 53946 (5 + 3 + 9) × 4 + 6
28740 2 × (8 + 7) × 4 + 0 • = . 53169 53 + 169
21960 2 × (1 + 9) + 60 = . 34587 34 + 5 + 87
29160 (29 + 1) × 6 + 0 • = . 35478 3 + (5 × 4 + 7) × 8
•
2 + 3 × 46 + 0 23460 = . 78591 7 × (8 + 59 × 1)
•
•
24190 2 × 41 × 9 + 0 = . 36875 3 × (68 + 7) × 5
•
•
2 × 45 × 7 + 0 24570 = . 38961 38 + 961
•
•
•
29370 2 × 9 + 37 + 0 = . 65148 6 × (5 + 14) + 8
36720 36 × 7 + 20 = . 49815 4 + (9 × 8 + 1) × 5
37590 3×7+5+9+0 • = . 81624 8 × (1 + 6 + 2) + 4
•
•
38190 38 × (1 + 9) + 0 = . 45627 4 × 5 + 62 × 7 3 + 85 + 7 + 0 38570 = . 49126 4 + 9 × (1 + 2 × 6)
31590 3×1×5×9+0 = . 76284 7 × 6 + 284
•
38610 (3 + 8) × 6 × 1 + 0 = . 49725 4 × 9 + 7 × (2 + 5)
•
31620 3 × 16 + 20 = . 89745 8 + (9 + 7 × 4) × 5
•
39240 3 × 92 + 4 + 0 = . 76518 7 × 6 × (5 + 1 × 8)
2 + (4 + 5) × 7 + 0 24570 = . 83916 8×3×9×1+6
•
31860 3 × 18 + 6 + 0 = . 47259 4 × 7 + 2 + 59
•
39410 3 + 94 + 1 + 0 = . 87265 87 + 2 × 65
•
48510 (4 + 8) × 5 + 10 = . 76923 (7 + (6 + 9) × 2) × 3
25160 25 + 1 × 60 = . 97384 9 + (7 + 3) × 8 × 4
•
31920 3 + 1 × 92 + 0 = . 56784 (5 + 6) × (7 + 8) + 4
39480 (3 + 9) × 4 + 8 + 0 = . 72615 7 × 2 × (6 + 1) + 5
•
•
48930 4 + 8 + 93 + 0 = . 51726 (5 × 17) + 26
•
49320 4 × 9 × (3 + 2) + 0 = . 51786 5 + 178 + 6
•
51240 5 × 1 × 24 + 0 = . 67893 67 + 89 + 3
•
51840 5 + 1 + 84 + 0 = . 92736 9 + 2 × 73 + 6
25830 2×5×8×3+0 • = . 69741 6 × 9 × (7 + 4 + 1)
34170 3 × (4 + 1) + 70 • = . 58692 (58 + 6 + 9) × 2
•
39840 3 + 9 + 8 + 40 = . 61752 6 + 17 × 5 + 2
34560 (3 × 4 × 5) + 60 = . 78912 7 + 89 × (1 + 2)
•
•
34860 34 × 8 + 60 = . 71295 7 × 1 × (2 + 95)
•
41580 4 × 15 + 80 = . 76329 76 × 3 + 29
•
51840 5 × 1 × (8 + 40) = . 97632 9 + 7 × 63 + 2
•
35 × 2 × 8 + 0 35280 = . 69174 6 × (9 + 174)
•
41580 (4 + 1 + 5) × 8 + 0 = . 76923 7 + 69 × 2 + 3
•
5 × (2 + 3) + 60 52360 = . 91784 9 + (17 × 8 + 4)
25930 2 × (5 × 9 + 30) = . 67418 6 + (7 + 41) × 8
•
•
26190 2 × (6 + 1 + 90) = . 38475 3 × (8 + 4 + 7) × 5
•
26580 (2 + 6) × 5 + 80 = . 49173 (4 + (9 + 1) × 7) × 3
•
36450 3 × (6 + 4) × 5 + 0 = . 71928 (7 + 1) × (9 + 28)
(2 + 7) × (5 + 40) 27540 = . 39168 (3 + 9 × 1) × 6 × 8
21580 2 × (1 + 5) + 8 + 0 = . 46397 4×6+3+9+7
•
•
41520 4 × 15 + 20 = . 78369 7 + 8 × (3 + 6 + 9)
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
52910 5 + 2 × 9 × 10 = . 78364 (7 + 8) × 3 × 6 + 4
•
71230 7 + 1 + 2 × 30 = . 98465 9 + 8 × (4 + 6) + 5
•
2190 2×1×9+0 2640 2×6+4+0 = . • = . 684375 (6 × (8 + 4) + 3) × 75 381975 (3 × 8 × 19 + 7) × 5
•
53460 534 + 6 + 0 = . 87219 872 + 1 × 9
•
7 × 5 × 2 + 40 75240 = . 81396 8 × 13 + 9 + 6
•
2340 2×3+4+0 = . 175968 (1 + 75) × 9 + 68
54270 5 × 4 × 27 + 0 • = . 68139 6 × (8 × 13 + 9)
•
54630 54 + 6 + 30 = . 92871 9 × (2 + 8 + 7 × 1)
82160 (8 + 2) × 16 + 0 • = . 93457 (9 + 3 × 4 + 5) × 7
•
8 × (4 + 16) + 0 84160 = . 95732 9 + 57 × 3 + 2 85410 (8 + 5) × (4 + 1) + 0 = . 97236 9×7+2+3+6
•
2670 2+6×7+0 = . 481935 4 + 81 × (93 + 5)
2790 • (2 + 3) × 4 + 0 2340 143685 = . • 185679 (18 + 5) × (6 + 7 × 9) 2810 • 439765 2340 2×3+4+0 • = . 185796 1 + (8 + 5) × (7 + 9 × 6) 2850 • 143697 2350 (2 + 3) × 5 + 0 • = . 2940 194768 (1 + 9 × 4 × 7 + 6) × 8 • 316785
=
27 + 9 + 0 . 14 + 368 × 5
=
2+8×1+0 . (4 + 3 × (97 + 6)) × 5
=
(2 + 8) × 5 + 0 . 1 + (4 + 36) × 9 × 7
=
(2 + 9) × 4 + 0 . 31 + 6 × 785
54720 5 × (4 × 7 + 20) = . 81396 (8 + 1) × 39 + 6
•
•
54780 5 + 47 + 8 + 0 = . 93126 (9 + (3 + 1) × 2) × 6
•
1230 1×2+3+0 = . 897654 89 × (7 + 6 × 5 + 4)
•
2+3+7+0 2370 2970 2+9+7+0 = . = . 946815 94 × (6 + (8 + 1) × 5) • 184635 184 × 6 + 3 × 5
•
5 × (6 + 4 × 9) + 0 56490 = . 71823 (7 + 1 × 82) × 3
•
1290 1 + 29 + 0 = . 768453 7 × (6 + 845) × 3
•
2430 2 + 43 + 0 = . 159786 1 + (5 × 97 + 8) × 6
58160 5×8×1×6+0 • = . 79243 (79 + 2) × 4 + 3
•
1×3+7+0 1370 = . 859264 (8 + 5 × 9 × 2) × 64
•
3190 3 + 19 + 0 2430 (2 + 4) × 3 + 0 • = . = . 248675 (24 × (8 + 6) + 7) × 5 186975 (18 × (6 + 9) + 7) × 5
58320 5 × 8 + 320 • = . 96714 9 + 6 × 7 × 14
•
1390 1×3+9+0 = . 642875 (64 + 2 + 8) × 75
•
•
59280 5 × (9 × 2 + 8) + 0 = . 73416 7 × (3 + 4 + 16)
•
1520 1×5×2+0 = . 438976 4 × (38 + 9 × 76) 1680 16 + 80 = . 249375 (2 + 4 × 9) × 375
•
2970 2 × (9 + 7 + 0) = . 461835 (4 + 618) × (3 + 5)
3190 • 2430 24 + 30 642785 • = . 198675 (1 + 9 × (8 + 6) × 7) × 5 3210 • 957864 2×4×3+0 2430 = . • 671895 6 × (71 + 8) × (9 + 5) 3240 • 158976 2430 2 + 4 + 30 • = . 679185 (6 + 7) × 9 × (1 + 85) 3240 • 176985
=
3+1×9+0 . 6 × (4 + 27) × (8 + 5)
=
3 + 2 + 10 . (9 × 5 + 7) × 86 + 4
=
3 + 2 + 40 . (1 + 5 × (8 × 9) + 7) × 6
=
32 × 4 + 0 . 1 × 7 + 6985
59840 5 × (98 + 4) + 0 = . 73216 (7 + 32) × 16
•
•
6 + 174 + 0 61740 = . 89523 ((8 + 9) × 5 + 2) × 3
•
1+6+8+0 1680 = . 395472 3 + (9 × 5 + 4) × 72
•
(2 + 4) × 3 + 0 2430 = . 871695 8 + 716 × 9 + 5
•
3240 3×2×4+0 = . 186975 (18 × (6 + 9) + 7) × 5
•
62730 6 × 2 + 73 + 0 = . 98154 9 + 8 × 15 + 4
•
1740 1×7×4+0 = . 239685 2 + (3 + 96 × 8) × 5
•
2470 2+4×7+0 = . 319865 (3 + 1 × 9 × 86) × 5
•
3240 3 × 24 + 0 = . 198675 1 + 9 × (8 + 6) × 7 × 5
•
63950 6 + 39 + 5 + 0 = . 74182 (7 + 4 + 18) × 2
•
1920 1 + 9 + 20 = . 576384 57 × (6 + 38 × 4)
•
2×4+8+0 2480 = . 316975 (31 + 6 × 9 × 7) × 5
•
3240 3 + 2 + 40 = . 571896 5 + 7 × 189 × 6
•
3240 3 × 2 × (4 + 0) = . 871695 8 + 716 × 9 + 5
•
3270 3 + 27 + 0 = . 416598 (4 + (1 + 6) × 5) × 98
•
3270 3 + 27 + 0 = . 954186 9 × 54 × 18 + 6
•
3290 3+2+9+0 = . 146875 1 + (4 + 6 × 8) × (7 + 5)
•
64320 64 × (3 + 2) + 0 • = . 71958 7 × (1 + 9) × 5 + 8
•
64870 6 × (4 × 8 + 7) + 0 = . 92315 9 × 2 + 315
1930 1+9×3+0 • = . 284675 (28 × 4 + 6) × 7 × 5
•
65930 6 + 59 + 30 = . 71482 7 + 1 × 48 × 2
•
•
68730 6×8+7+3+0 = . 91245 9×1×2×4+5
•
69240 (6 + 9) × 2 × 4 + 0 = . 81357 81 + 3 + 57
•
2480 24 + 8 + 0 • = . 319765 31 + 9 × 7 × 65
2610 2×6×1+0 2 + 13 + 0 2130 •. = . = 384975 (3 + 8 + 49 × 7) × 5 497568 (4 × 9 × (7 + 5) + 6) × 8 2610 2 × (6 + 1 + 0) = . 974835 9 × 7 × (48 + 35)
2140 (2 + 1) × 4 + 0 = . 869375 (8 + 6 × 9 + 3) × 75
•
•
2180 2 + 18 + 0 = . 765943 7 + 65 × 9 × 4 × 3
•
2640 2×6+4+0 = . 34 + 2 + 0 183975 (1 × 8 × 3 × 9 + 7) × 5 • 3420 = . 619875 (6 + 1 × 9) × 87 × 5
•
2180 2 × 18 + 0 = . 934675 9 × (3 + 46) × 7 × 5
•
2 + 6 + 40 2640 = . 193875 1 × (9 + 38) × 75
•
3510 3 × (5 + 10) = . 248976 2 × (4 + 8 + 9) × 76
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
3510 3×5×1+0 4260 4 + 26 + 0 4860 4+8×6+0 = •. = •. = . 279864 (2 × 7 + 9) × (8 × 6 + 4) 137598 1 × (3 × (7 × 5 × 9 + 8)) 139725 13 × (9 + 7 × 2) × 5
•
5460 5 × 46 + 0 = . 192738 1 + ((9 + 2) × 738)
•
3510 3 × (5 + 10) = . 297648 (29 + 7 × 64) × 8
•
•
5480 5 × (4 + 8) + 0 = . 196732 1 + 9 + 67 × 32
3×5×1+0 3510 = . 482976 (4 × 82 + 9 + 7) × 6
4 + 8 + 60 4860 4320 4 × 3 + 20 = . • = . • 197235 1 + 972 × 3 + 5 186975 (18 × (6 + 9) + 7) × 5
•
•
5490 5×4×9+0 = . 127368 (12 × 7 + 3) × 6 × 8
•
5 + 670 5670 = . 123984 (12 + 3) × 984
•
5670 5 × 6 + 70 = . 139482 (13 × 94 + 8) × 2
•
5670 5 × (6 + 7) + 0 = . 312984 3 × (1 + 298) × 4
•
5740 5 × 7 + 40 = . 231896 (2 × 31 × 8 + 9) × 6
•
5810 5×8×1+0 = . 349762 (34 + 9) × 7 × (6 + 2)
5×1×3+0 5130 = . 248976 2 × (4 × 8 × 9 + 76)
•
5 × (9 + 1) + 0 5910 = . 238764 2×3×8×7×6+4
3510 3×5×1+0 • = . 897624 (897 + 62) × 4
•
•
3510 3×5×1+0 = . 924768 (9 × 2 + 476) × 8 3+5+7+0 3570 = . 184926 (1 + 84) × 9 + 2 × 6
4310 4 × (3 + 1 + 0) 4860 4 + 86 + 0 = .• = . 829675 8 × (2 + 9 × 6 × 7 + 5) 175932 1 + 7 × 5 × 93 + 2
48 + 60 4860 4320 4 + 3 × 20 • = . • 291375 = (2 × 91 + 3) × 7 × 5 . 679185 (6 + 7) × 9 × (1 + 85)
•
•
4320 4 × 3 + 20 = . 871695 8 + 716 × 9 + 5
•
4510 4+5+1+0 = . 283679 2 + 8 × 3 + 67 × 9
•
4 × 5 + 10 4510 = . 782936 7 × 8 × (29 × 3 + 6)
•
4510 4+5+1+0 = . 876293 8 × 7 + 629 × 3
•
4860 48 + 60 = . 739125 73 × 9 × 1 × 25 4870 (4 + 8) × 7 + 0 = . 192365 1 + 92 × 36 + 5 4920 4+9+2+0 = . 358176 (3 × 58 + 1 + 7) × 6
3690 3 × (6 + 9) + 0 = . 271584 (271 + 5) × (8 + 4)
•
•
3720 3 + 7 + 20 = . 156984 1 + 5 + (6 + 9) × 84
•
•
4530 4×5×3+0 3760 3+7+6+0 •. = . = (2 + 9 + 71) × 6 × 8 129485 1 + (2 + 9 × (4 + 8)) × 5 297168
•
5130 5 × (1 + 3 + 0) = . 642789 6 + 4 × (2 + 7 × 89)
•
6120 6+1×2+0 = . 384795 3 + (8 + 47) × 9 + 5
•
3×7×8+0 3780 = . 169425 (16 × 94 + 2) × 5
•
4620 (4 + 6) × 2 + 0 = . 193578 1 + 9 × (3 × 5 + 78)
•
5230 5×2×3+0 = . 719648 (71 + 9 + 6) × 48
•
6+1×2+0 6120 = . 947835 9 + (4 + 78) × 3 × 5
•
3780 3 + 7 + 80 = . 261954 (2 + 61) × (95 + 4)
•
4630 4 × (6 + 3 + 0) = . 891275 (8 + 91) × 2 × 7 × 5
•
5 × 24 + 0 5240 = . 173968 (1 + 73 + 9) × 6 × 8
•
6180 (6 + 1) × 8 + 0 = . 239475 2 × (3 × 9 + 4) × 7 × 5
•
6180 (6 + 1) × 8 + 0 = . 749325 7 × (4 + 93) × 2 × 5
•
6210 6×2×1+0 = . 379845 3 + 7 × 98 + 45
•
6370 63 + 7 + 0 = . 249158 (2 + 4) × 91 × 5 + 8
•
6390 6 + 39 + 0 = . 158472 (15 × 8 + 4) × (7 + 2)
•
6420 (6 + 4) × 2 + 0 = . 375891 (3 + 7) × (5 + 8) × 9 + 1
•
6450 (6 + 4) × 5 + 0 = . 127839 1 + (27 + 83) × 9
•
3 + 87 + 0 3870 = . • 126549 1 + (2 + 6 × 54) × 9
4680 4 × 6 + 80 • = . 192375 (1 + 9 × 2) × 3 × 75
3940 (3 + 9) × 4 + 0 • = . 217685 2 × 17 × 6 × (8 + 5)
5280 5 + 2 × 80 4680 4×6+8+0 • = . • 149376 = 14 × 9 × 37 + 6 . 219375 (21 + 9) × (3 + 7) × 5
•
•
•
4150 (4 + 1) × 5 + 0 = . 297638 297 × 6 + 3 + 8 4150 (4 + 1) × 5 + 0 = . 896732 8 × (96 × 7 + 3) + 2 4230 4+2×3+0 = . 185697 (18 + 5 × 6) × 9 + 7
•
•
•
4680 4 × 6 + 80 = . 291375 (2 × 91 + 3) × 7 × 5
5240 5 × 24 + 0 • = . 179863 1 × 7 × 98 × 6 + 3
•
4680 4 × 6 + 80 = . 739125 73 × 9 × 1 × 25
•
4760 4 + 76 + 0 = . 139825 1 + 3 × 9 × (82 + 5)
•
5310 5 × (3 + 1 + 0) = . 427986 4 + 2 × (798 + 6) 5 × (3 + 1 + 0) 5310 = . 786942 78 × (6 + 9 + 4) × 2 5370 5+3+7+0 = . 149286 1 + 4 × 92 + 8 × 6
•
4230 4+2×3+0 5370 5+3+7+0 4830 4 + 8 + 30 = . • = . 571896 (5 + 7 + 1) × (8 + 96) • 217695 = 2 + 1 + 7 × 6 × 9 × 5 . 249168 (24 + 9 × (1 + 6)) × 8 • 6480 = (6 + 4) × 8 + 0 . 179253 1 + 79 × (25 + 3)
•
4230 4 × (2 + 3 + 0) = . 768591 7 × (6 × 85 + 9) + 1
•
4830 4 + 8 + 30 = . 219765 21 + 9 × 7 × 6 × 5
•
5410 5+4+1+0 = . 836927 8 + 3 × (69 + 2) × 7
•
6480 (6 + 4) × 8 + 0 = . 193752 1 × (9 + 37) × 52
•
4230 4 + 2 × (3 + 0) = . 791856 7 + 9 + 1856
•
4860 4 + 86 + 0 = . 137592 13 × 7 × (5 + 9) × 2
•
5430 5 × (4 + 3) + 0 = . 186792 1 × 8 + (6 + 7) × 92
•
6480 64 + 80 = . 291375 (2 × 91 + 3) × 7 × 5
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
6480 (6 + 4) × 8 + 0 = . 357291 35 × 7 × 2 × 9 + 1
•
(7 + 3) × 5 + 0 8+1+7+0 7350 8170 = = . •. 291648 (2 + (9 + 1) × 6) × 4 × 8 396245 3 × (9 + 62 × 4) + 5
•
6480 6 + 4 + 80 = . 715392 (7 × 15 + 3) × 92
•
7380 (7 + 3) × 8 + 0 = . 561249 (56 × 12 + 4) × 9
•
6510 6×5×1+0 7410 7 × 4 + 10 = . • = . 938742 ((9 + 3) × 8 + 7) × 42 386295 3 + 86 × (2 × 9 + 5)
•
6720 6+7×2+0 = . 359184 3 + 59 × 18 + 4
•
8 × 3 + 10 8310 6840 (6 + 8) × 4 + 0 7490 7+4+9+0 = .• = . • 976425 = (9 + 76) × (42 + 5) . 123975 1 × (2 + 3 × 9) × 7 × 5 653128 (6 × (5 + 31) + 2) × 8
•
6970 6 + 9 + 70 = . 143582 1 + 4 + 3 × 582
7×5×6+0 7560 = . 123984 1 × (2 + 39) × 84
•
•
7140 7×1×4+0 7560 7×5×6+0 = . • = . 236895 2 + (3 + 6) × (8 + 95) 198324 1 + (9 + 8) × 324
•
7140 71 + 4 + 0 7560 75 + 60 = . • = . 369852 (3 + 69 × 8) × (5 + 2) 419328 (4 + 1 × 932) × 8
•
7140 7×1×4+0 = . 896325 (89 + 6) × (32 + 5)
•
•
7280 7 × (2 + 8) + 0 = . 419536 4 × 19 × 53 + 6
•
7430 74 + 30 = . 962185 962 × (1 + 8 + 5)
•
8640 8 × (6 + 4 + 0) = . 715932 7 × (15 + 932)
•
8640 8 × (6 × 4 + 0) = . 739125 73 × 9 × 1 × 25
•
8190 8 × (1 + 90) = . 374625 3 × 74 × 6 × 25
•
8230 (8 + 2) × 3 + 0 8670 8 + 67 + 0 = . • = . 754691 7 × (5 + 4 × (6 + 91)) 521934 5 × 21 × (9 + 34)
•
8310 8×3×1+0 = . 427965 4 × (279 + 6 × 5)
•
8720 8 + 72 + 0 = . 659341 65 × 93 + 4 × 1
•
8730 8 × (7 + 3 + 0) = . 649512 6 × (495 + 1) × 2
8340 8+3+4+0 = . 156792 156 + 7 × 9 × 2
•
8760 8 × (7 + 6) + 0 = . 132495 1 × 32 × 49 + 5
•
8370 83 + 7 + 0 = . 126945 1 × (269 + 4) × 5
•
9180 (9 + 1) × 8 + 0 = . 375462 3 + 7 × (5 + 462)
•
8370 8 × (3 + 7) + 0 = . 246915 (2 + 469 + 1) × 5
•
(9 + 1) × 8 + 0 9180 = . 567324 (5 + 67 × 3) × 24
•
8370 8 × (3 + 7) + 0 = . 649512 64 × (95 + 1 × 2)
•
9240 9+2+4+0 = . 317856 (3 + 1 × 78 + 5) × 6
7820 78 + 2 + 0 7180 7+1+8+0 •. = . = 1 + 3 × (6 + 459) 493625 (4 + 9 × 3 × (6 + 2)) × 5 136459
•
8370 8 × 3 + 70 = . 941625 9 × (41 + 6) × 25
•
9270 9×2+7+0 = . 541368 5 × (4 + 1 × 36 × 8)
•
7 + 83 + 0 7830 = . 129456 12 × (94 + 5 × 6)
•
8460 8+4×6+0 9+2+7+0 9270 = . 391275 (39 + 1) × (2 + 7 × 5) • 681345 = (6 × 8 + 1) × 3 × (4 + 5) .
7290 72 + 90 = . 143865 14 × 38 × 6 + 5
•
7830 7+8+3+0 = . 264915 (2 + 64) × 9 + 15
•
8640 8 × (6 + 4) + 0 = . 153792 (15 + 3) × 79 + 2
•
9360 9 + 36 + 0 = . 275184 27 × (5 × (1 + 8) + 4)
•
7 + 29 + 0 7290 = . 148635 14 + 8 × 6 × 3 × 5
•
7830 7 + 83 + 0 = . 514692 51 × (4 + 6 × 9) × 2
•
8640 8×6×4+0 = . 192375 (1 + 9 × 2) × 3 × 75
•
9460 94 + 6 + 0 = . 135278 13 × 5 × (2 × 7 + 8)
•
7290 72 + 90 = . 168345 1 × 6 + 83 × 45
•
8640 8×6×4+0 7+8+9+0 7890 = . = . • 291375 (2 × 91 + 3) × 7 × 5 264315 2 × (64 + 3) × (1 + 5)
•
9540 9 × (5 + 40) = . 173628 1 + 7362 + 8
•
7320 (7 + 3) × 2 + 0 8130 8 + 1 × 30 = •. = . 169458 (1 + 6) × (9 + 4) × 5 + 8 296745 2 + (9 + 67 × 4) × 5
•
9810 9+8+1+0 = . 426735 4 × 2 × 6 + 735
2.5
•
•
•
7590 75 + 90 = . 183264 1 × 83 × 2 × 6 × 4
•
8640 8 × (6 + 4) + 0 = . 319572 31 × 95 + 7 × 2
Fractions Ending in Zero in Denominator
The above subsection 2.4 brings results having 0 in the numerator, but this subsection brings fractions ending in 0 in the denominator. Again, dividing by 10 don’t guarantee the existence of selfie fraction as in case of subsection 2.1.
135 1+3+5 = . 480 4×8+0
•
•
165 1×6+5 = . 480 4×8+0
•
168 1 × (6 + 8) = . 540 5 + 40
•
174 1+7+4 = . 580 5×8+0
•
319 3 + 19 = . 580 5×8+0
•
195 1×9+5 = . 780 7×8+0
•
392 3+9×2 = . 560 5×6+0
•
245 2 × (4 + 5) = . 980 9×8+0
•
396 3×9+6 = . 540 5 + 40
•
425 (4 + 2) × 5 = . 680 6×8+0
•
135 1×3+5 = . 8640 8 × 64 + 0
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
135 1+3+5 = . 9720 9 × 72 + 0
•
715 7 + 15 = . 8320 8 × 32 + 0
•
1485 14 + 85 = . 9720 9 × 72 + 0
•
2485 2 + 4 + 85 = . 6390 6 × 39 + 0
•
138 1+3+8 = . 5290 5 × (2 + 90)
•
725 (7 + 2) × 5 = . 9860 9 × (8 + 60)
•
1584 1+5+8+4 = . 3960 3 × (9 + 6) + 0
•
2496 2 × (4 + 9) + 6 = . 3510 3 × (5 + 10)
•
142 1×4+2 = . 5680 5×6×8+0
•
782 7+8×2 = . 1530 15 + 30
•
1682 1 + (6 + 8) × 2 = . 9570 95 + 70
•
2695 2 + (6 + 9) × 5 = . 7840 7×8×4+0
•
1×4+5 145 = . 9860 9 × (8 + 60)
•
798 7+9+8 = . 5320 5 × 32 + 0
•
1683 1 × 6 × (8 + 3) = . 4590 4×5×9+0
•
2816 28 × 16 = . 5940 5 + 940
•
165 1×6+5 = . 9720 9 × 72 + 0
•
8+7+3 873 = . 1940 (1 + 9) × 4 + 0
•
1×6×8+3 1683 = . 5940 5×9×4+0
•
2871 2 × 8 + 71 = . 5940 5×9×4+0
•
(2 + 1) × 3 213 = . 5680 5×6×8+0
•
918 9+1×8 = . 2430 2 + 43 + 0
•
1734 1+7+3+4 = . 2890 2×8+9+0
•
2 × (9 + 6 + 4) 2964 = . 3510 3 × (5 + 10)
•
214 2+1×4 = . 8560 8×5×6+0
•
918 9 + 18 = . 3570 3×5×7+0
•
1736 17 + 3 × 6 = . 2480 2 + 48 + 0
•
(31 + 2) × 4 3124 = . 5680 5×6×8+0
•
2×3+1 231 = . 5940 5×9×4+0
•
9×2+4 924 = . 3570 3 × 5 + 70
•
1758 17 + 5 × 8 = . 2930 2 + 93 + 0
•
3159 3 × 15 + 9 = . 4680 (4 + 6) × 8 + 0
•
243 2+4+3 = . 5670 5×6×7+0
•
9×2+4 924 = . 3780 3 + 7 + 80
•
1845 (1 + 8) × 4 + 5 = . 7290 72 + 90
•
3168 31 + 68 = . 5920 5 + 9 × 20
•
3+1+5 315 = . 7840 7×8×4+0
•
946 9+4×6 = . 2580 2 × 5 + 80
•
1863 18 × (6 + 3) = . 5290 5 × (2 + 90)
•
3168 3 × 168 = . 5940 5 + 940
•
321 3 × (2 + 1) = . 8560 8×5×6+0
•
981 9+8+1 = . 6540 6×5×4+0
•
1925 1 × (9 + 2) × 5 = . 7840 7×8×4+0
•
3192 3 × (1 + 9 × 2) = . 7560 75 + 60
•
324 3+2+4 = . 7560 7×5×6+0
•
984 9+8×4 = . 1560 1 × 5 + 60
•
1936 1×9×3+6 = . 5280 5 × 2 + 80
•
3 × (26 + 7) 3267 = . 5940 5×9×4+0
•
3×2+7 327 = . 6540 65 × 4 + 0
•
1265 1 + 2 × (6 + 5) = . 8470 84 + 70
•
1938 1+9+3×8 = . 4560 4 × 5 + 60
•
3465 3×4×6+5 = . 7290 72 + 90
•
345 3+4×5 = . 9720 9 × 72 + 0
•
1365 1+3×6×5 = . 9720 9 × 72 + 0
•
2178 2 + (1 + 7) × 8 = . 5940 5×9×4+0
•
3 × (4 + 6) + 5 3465 = . 8910 89 + 1 + 0
•
352 (3 + 5) × 2 = . 1980 1 + 9 + 80
•
1376 1 × 37 + 6 = . 5280 5 + 2 × 80
•
2184 2 × 1 × (8 + 4) = . 6370 63 + 7 + 0
•
3485 3 × (4 + 8) + 5 = . 6120 6 × 12 + 0
•
546 5×4+6 = . 1890 1 + 89 + 0
•
1 × 37 + 6 1376 = . 5920 5 + 9 × 20
•
(2 + 3) × 5 + 6 2356 = . 7980 7 + 98 + 0
•
3564 3+5+6×4 = . 8910 8 × (9 + 1) + 0
•
592 5+9+2 = . 1480 (1 + 4) × 8 + 0
•
1386 1 × 3 × (8 + 6) = . 5940 5×9×4+0
•
2358 2+3+5+8 = . 9170 (9 + 1) × 7 + 0
•
3572 3 × (5 + 7) + 2 = . 8460 84 + 6 + 0
•
612 6 + 12 = . 3570 3×5×7+0
•
1452 1 × (4 + 5) × 2 = . 9680 (9 + 6) × 8 + 0
•
2415 2 + 41 × 5 = . 6790 6 × (7 + 90)
•
3645 36 × (4 + 5) = . 8190 8 × (1 + 90)
•
627 6×2+7 = . 5940 5×9×4+0
•
1462 1+4+6×2 = . 3870 3 × (8 + 7) + 0
•
2415 2 + 41 × 5 = . 8960 8 × 96 + 0
•
3648 3 × (6 + 4 × 8) = . 5920 5 + 9 × 20
•
6+2+9 629 = . 1480 (1 + 4) × 8 + 0
•
1485 1+4+8+5 = . 6930 6 × 9 + 30
•
2475 2×4+7+5 = . 8910 8×9×1+0
•
3796 37 + 9 + 6 = . 5840 5 × 8 + 40
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
3854 3+8×5+4 = . 6970 6 + 9 + 70
•
6435 (6 + 4 + 3) × 5 = . 8910 89 + 1 + 0
•
396 3+9+6 = . 15840 15 × (8 + 40)
•
4125 (4 + 1) × 2 × 5 = . 6930 6 × 9 + 30
•
6534 6+5×3×4 = . 8910 89 + 1 + 0
•
412 4 × (1 + 2) = . 57680 5×7×6×8+0
•
4192 4 × (1 + 9 + 2) = . 7860 (7 + 8) × 6 + 0
•
6×7×3+1 6731 = . 9540 9×5×4+0
•
4257 4×2+5×7 = . 8910 89 + 1 + 0
•
7 + 3 + 26 7326 = . 8140 8 × (1 + 4) + 0
•
4 × (5 + 1) × 2 4512 = . 9870 98 + 7 + 0
•
7395 7 × 3 + 95 = . 8160 8 × 16 + 0
4653 4 × (6 + 5) + 3 = . 8910 89 + 1 + 0
•
•
7425 (7 + 4 × 2) × 5 = . 8910 89 + 1 + 0
4732 4 × (7 + 3 × 2) • = . 8190 81 + 9 + 0
74 + 6 + 1 7461 = . • 8290 (8 + 2) × 9 + 0
4761 4 + 76 + 1 • = . 5290 5 × (2 × 9 + 0)
7623 7 × (6 + 2 + 3) • = . 8910 89 + 1 + 0
•
•
•
•
•
•
4895 4 + (8 + 9) × 5 = . 7260 72 + 60 5236 5 × (2 + 3 + 6) = . 7140 71 + 4 + 0 5392 (5 + 3 × 9) × 2 = . 6740 6 + 74 + 0 5742 5 + (7 + 4) × 2 = . 6380 6+3×8+0 5823 (5 + 8 × 2) × 3 = . 6470 (6 + 4) × 7 + 0 5826 5×8+2+6 = . 9710 9 + 71 + 0
•
936 9+3+6 = . 25480 2 × 5 + 480
•
936 9+3×6 = . 51480 5 + 1480
•
936 9+3×6 = . 58240 5 × 8 × (2 + 40)
•
986 9+8+6 = . 54230 5 + 42 × 30
•
1×2×4+3 1243 = . 58760 5 × 8 × (7 + 6) + 0
•
1274 1 + 2 + 74 = . 53690 5 + 36 × 90
•
1275 1 × (2 + 7) × 5 = . 48960 4×8×9×6+0
542 (5 + 4) × 2 = . 18970 (1 + 89) × 7 + 0
•
1295 (1 + 2) × (9 + 5) = . 34780 3 × 47 × 8 + 0
•
6+1+2 612 = . 39780 39 × (7 + 8) + 0
•
1×3+2+8 1328 = . 59760 5 × 9 × (7 + 6) + 0
•
627 6×2+7 = . 15840 15 × 8 × 4 + 0
•
1428 1×4+2+8 = . 59670 5 × 9 × (6 + 7) + 0
•
7×1×4 714 = . 29580 29 × 5 × 8 + 0
•
1428 1 + 42 + 8 = . 95760 9 × 5 × 76 + 0
•
715 7 + 15 = . 24960 2 × 4 × 96 + 0
•
1+4+3+7 1437 = . 52690 5 × 2 + 6 × 90
•
7 + 26 726 = . 15840 15 × (8 + 40)
•
1473 14 + 7 + 3 = . 58920 5 × 8 + 920
•
(1 + 4) × 7 + 6 1476 = . 35280 35 × 28 + 0
•
1476 1+4+7+6 = . 38950 3 + 8 × (9 + 50)
•
1485 14 + 85 = . 23790 2 × (3 + 790)
•
1485 1+4+8+5 = . 37620 3 × 76 × 2 + 0
864 8+6+4 = . 17520 1 + 7 × 52 + 0
•
1624 (1 + 6) × 2 × 4 = . 38570 38 × 5 × 7 + 0
4+1×3 413 = . • 57820 5 × 7 × (8 + 20) 421 4×2×1 • = . 37890 (3 + 7) × 8 × 9 + 0
•
•
•
•
•
•
•
•
•
8 + 2 + 53 8253 = . 9170 (9 + 1) × 7 + 0 135 1+3+5 = . 64890 6 + 48 × 90 1×6+5 165 = . 23790 2 × (3 + 790) 192 1+9+2 = . 45360 45 × (3 + 60) 231 2×3+1 = . 78540 7 × 85 × 4 + 0 231 2 × (3 + 1) = . 85470 8 × 5 × (4 + 70)
431 4+3+1 = . 25860 2×5×8×6+0 461 4 × (6 + 1) = . 87590 8 × 7 × (5 + 90)
754 7+5+4 • = . 26390 2 + 6 × (3 + 90)
5916 5 + 9 × (1 + 6) • = . 7830 7 + 83 + 0
234 2+3+4 • = . 19760 (1 + 9) × 76 + 0
•
6125 (6 + 1) × 25 = . • 7840 7×8×4+0
235 2 × (3 + 5) • = . 78960 7 × 8 × 96 + 0
•
7+6+5 765 = . 19380 19 × 3 × 8 + 0 792 7+9+2 = . 53460 5 × (3 + 4 × 60)
•
6312 (6 + 3) × 12 = . 7890 (7 + 8) × 9 + 0
•
261 2+6+1 = . 38570 38 × 5 × 7 + 0
•
•
6345 6 + 3 × 45 = . 7290 72 + 90
•
315 3+1×5 = . 86940 8 × 69 × 4 + 0
•
891 8+9×1 = . 53460 5 × 34 × 6 + 0
•
1634 1+6+3×4 = . 27950 (2 + 7 × 9) × 5 + 0
•
64 + 35 6435 = . 7280 7×2×8+0
•
316 3+1+6 = . 75840 75 × 8 × 4 + 0
•
935 9+3+5 = . 12870 (1 + 2) × (8 + 70)
•
1725 1+7×2+5 = . 49680 496 + 80
•
6435 6×4×3+5 = . 8190 8 + 1 × 90
•
3+2+5 325 = . 17680 (1 + 7) × 68 + 0
•
9+3×5 935 = . 48620 48 × (6 + 20)
•
1+7+2+5 1725 = . 63480 6 × (3 × 4 + 80)
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
1728 1×7×2×8 = . 53460 5 + 3460
•
2538 25 + 3 + 8 = . 19740 (1 + 9) × 7 × 4 + 0
•
3416 3 × (4 + 1 × 6) = . 59780 5 × (97 + 8) + 0
•
3976 3×9+7+6 = . 24850 (2 + 48) × 5 + 0
•
1734 17 + 34 = . 28560 28 × 5 × 6 + 0
•
2618 2×6+1+8 = . 95370 9 × (5 × 3 + 70)
•
3417 3 + 41 + 7 = . 56280 5 × 6 × 28 + 0
•
4128 (4 + 1 × 2) × 8 = . 96750 (9 + 6) × 75 + 0
•
1+7+4+6 1746 = . 35890 35 × 8 + 90
•
2673 26 + 7 + 3 = . 14850 (1 + 4) × 8 × 5 + 0
•
3×4+6×5 3465 = . 12870 1 × 2 × (8 + 70)
•
4173 4 + 1 + 73 = . 25680 2×5×6×8+0
•
1785 1+7+8+5 = . 23460 2 × 3 × 46 + 0
2+6+7+5 2675 = . 14980 (1 + 4 + 9) × 8 + 0
3465 3 × (4 + 6) + 5 • = . 17820 178 + 2 + 0
•
4+1+7+9 4179 = . 35820 3 × (58 + 2) + 0
•
4182 41 + 8 + 2 = . 35670 3 × 5 + 6 × 70
•
4185 4 + 18 + 5 = . 29760 2 × (9 + 7) × 6 + 0
•
4185 4 + 18 + 5 = . 36270 3 × (6 + 2 + 70)
•
4296 4 + 2 × (9 + 6) = . 75180 7 × (5 + 1 × 80)
3 × 6 × (4 + 5) 3645 = . 12870 12 + 8 × 70
•
43 + 6 × 8 4368 = . 17520 1 + 7 × 52 + 0
•
4521 4 × (5 × 2 + 1) = . 73980 (7 + 3) × 9 × 8 + 0
•
•
1785 1+7+8+5 = . 32640 3 × 2 × 64 + 0 1794 (17 + 9) × 4 = . 86250 8 × 625 + 0
•
•
•
1925 1 × (9 + 2) × 5 • = . 38640 3 × 8 × (6 + 40)
•
•
2975 2+9×7+5 = . 34680 3 × 4 × 68 + 0
•
3645 3×6+4+5 = . 17280 (1 + 7) × 2 × 8 + 0
•
2+9×7+5 2975 = . 43860 4 × 3 × 86 + 0
•
3654 3+6+5×4 = . 71820 71 × 8 + 2 + 0
•
4528 4 × (5 × 2 + 8) = . 36790 3 + 6 × (7 + 90)
•
3129 31 + 2 + 9 = . 47680 (4 + 76) × 8 + 0
•
3658 3×6+5+8 = . 19470 1 + 94 + 70
•
4532 4+5×3×2 = . 67980 6+7×9×8+0
•
3149 31 + 4 × 9 = . 28670 2 + 8 × (6 + 70)
•
3+7×2+4 3724 = . 15960 (1 + 5 + 9) × 6 + 0
•
4627 (4 + 6 + 2) × 7 = . 85930 8 × 5 × (9 + 30)
•
3182 (3 + 1 + 8) × 2 = . 95460 9 × (5 × 4 + 60)
3726 3+7×2+6 • = . 14580 1 + 4 + 5 + 80
•
4 × (6 + 5) + 3 4653 = . 17820 178 + 2 + 0
•
4693 (4 + 6 + 9) × 3 = . 51870 5 × 18 × 7 + 0
•
4695 (4 + 6 + 9) × 5 = . 28170 2 + 8 × (1 + 70)
•
4725 4×7+2+5 = . 18630 18 × 6 + 30
3816 3×8×1×6 = . 25970 2 × 5 + 970
•
4736 4 × (7 + 3 + 6) = . 12580 1 × 2 × (5 + 80)
2 × 1 × (6 + 5) 2165 = . • 38970 389 + 7 + 0 2168 2 × 1 × (6 + 8) • = . 37940 (3 + 7) × (9 + 40)
•
•
2175 2 × (1 + 7) × 5 = . 46980 4×6×9×8+0 2178 2×1×7+8 = . 35640 3×5×6×4+0 2178 2×1×7+8 = . 53460 534 + 6 + 0 2+1×8×5 2185 = . 34960 (3 + 4) × 96 + 0 2317 2×3+1+7 = . 49650 4 × (9 + 6) × 5 + 0
2376 2 × (3 + 7 + 6) = . • 14850 (1 + 4) × 8 × 5 + 0
3627 3 × (6 + 2) + 7 = . 51480 (51 + 4) × 8 + 0
2 × (9 + 5 × 6) 2956 = . 14780 (1 + 4) × 78 + 0
•
•
•
3625 3×6+2+5 = . 14790 1 + 4 + 7 + 90
•
1962 1×9×6×2 • = . 45780 45 × 7 × 8 + 0
•
2795 2 + 79 + 5 = . 16380 1 × 63 × 8 + 0
•
2945 2+9+4×5 = . 13680 1×3×6×8+0
•
•
2716 2 × (7 + 1 + 6) = . 35890 35 × 8 + 90
3195 3 + 19 + 5 = . • 62480 6 × (2 × 4 + 80)
•
3247 3×2+4+7 = . 91680 (9 + 1) × 6 × 8 + 0
•
•
3642 3 × (6 + 4 + 2) = . 78910 78 × (9 + 1) + 0
3752 3×7+5+2 = . 96480 (9 × 6 + 4) × 8 + 0 3795 3×7+9×5 = . 14260 1 × 4 × (2 + 60)
3249 3 × (2 + 4) + 9 = . 75810 7 × 5 × (8 + 10)
•
•
3267 3 × (26 + 7) = . 15840 15 × 8 × 4 + 0
•
3816 3+8+1+6 = . 57240 5 × (7 × 2 + 40)
•
4765 4 × (7 + 6) + 5 = . 38120 38 × 12 + 0
2478 2+4+7+8 = . 15930 1×5×9×3+0
•
3297 3+2+9+7 = . 81640 8 × (1 + 64) + 0
•
3876 3 × (8 + 7) + 6 = . 12540 125 + 40
•
4816 4 × (8 + 1 × 6) = . 27950 (2 + 7 × 9) × 5 + 0
2 + 5 × (1 + 6) 2516 = . 39780 39 × (7 + 8) + 0
•
(3 + 2) × (9 + 8) 3298 = . 17460 (1 + 74) × 6 + 0
•
3×8+9+5 3895 = . 74620 7 × 4 × (6 + 20)
•
48 + 2 × 9 4829 = . 17560 (1 + 7) × 5 × 6 + 0
•
2465 2 × (4 × 6 + 5) = . 19380 19 × 3 × 8 + 0
•
•
2465 2 × (4 × 6 + 5) = . 39780 (3 + 9) × 78 + 0
•
•
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
4896 48 + 9 × 6 = . 17520 1 + 7 × 52 + 0
•
5814 5+8+1×4 = . 27360 2 × (7 + 3) + 60
•
6795 6+7+9+5 = . 48320 4×8×3×2+0
•
7425 (7 + 4 + 2) × 5 = . 83160 8 × (31 + 60)
•
4896 48 + 9 + 6 = . 57120 5 × (7 × (1 + 20))
•
5842 5×8+4+2 = . 13970 1 + 39 + 70
•
6912 69 + 12 = . 35840 35 × (8 + 4) + 0
•
7429 74 + 2 × 9 = . 51680 5 × 16 × 8 + 0
•
4925 4+9+2+5 = . 76830 (7 + 6) × 8 × 3 + 0
•
5916 5 + 9 × (1 + 6) = . 27840 (2 + 78) × 4 + 0
•
6935 6+9×3+5 = . 28470 2 + 84 + 70
•
7429 7 + 4 × (2 + 9) = . 56810 5 × (68 + 10)
4983 4 × (9 + 8 × 3) • = . 25670 2 × 5 + 670
•
5918 5+9+1×8 = . 72630 7 + 263 + 0
6972 69 + 7 × 2 • = . 41580 415 + 80
•
(7 + 4) × 6 × 9 7469 = . 13580 135 × 8 + 0
61 × 3 + 7 6137 = . 25840 25 × 8 × 4 + 0
•
5146 5 × (1 + 4) + 6 = . 27390 2 + 73 + 90
•
7518 7+5+1+8 = . 42960 4 × 2 × (9 + 6) + 0
6175 (6 + 1 × 7) × 5 = . 32490 3 × (24 + 90)
•
5148 5 × (1 + 4) + 8 = . 32760 (3 + 2) × 7 × 6 + 0
•
7581 7+5+8+1 = . 32490 (3 × 2 + 4) × 9 + 0
•
7×5+8×2 7582 = . 69130 (6 + 9) × (1 + 30)
•
7614 7×6+1+4 = . 58320 5 × 8 + 320
7 + 1 × 34 7134 = . 29580 2 × 9 × 5 + 80
•
7623 7 × (6 + 2 + 3) = . 15840 1×5×8×4+0
•
7195 7 × (19 + 5) = . 86340 8 × 63 × 4 + 0
•
7685 7+6+8×5 = . 13920 1 + 3 + 92 + 0
•
7215 7 + 2 × 15 = . 43680 4 × 36 + 80
•
7685 7+6+8×5 = . 32190 3 + 219 + 0
•
7216 72 + 16 = . 38950 3 + 8 × (9 + 50)
•
7695 7+6+9+5 = . 14820 1 × 4 × 8 + 20
•
7×2+4+5 7245 = . 13860 1 × 38 + 6 + 0
•
7 + 8 × (1 + 2) 7812 = . 45360 4 × 5 × (3 + 6) + 0
7248 (7 + 2 × 4) × 8 • = . 19630 1 + 9 × (6 + 30)
•
7914 7 + 9 + 14 = . 26380 2 + 6 × 3 + 80
•
7931 79 + 31 = . 86520 8 × 6 × (5 + 20)
•
7×9+8+6 7986 = . 14520 14 × 5 × 2 + 0
•
8172 8 + 17 + 2 = . 95340 9 × 5 × (3 + 4) + 0
•
8192 8 × (1 + 9 + 2) = . 34560 345 + 60
•
•
•
•
•
5187 5+1+8+7 = . 29640 2 × (9 + 6) × 4 + 0 5 + 2 × 14 5214 = . 87690 8 + 7 + 6 × 90 5216 5×2×1+6 = . 37490 3 × 7 + 4 + 90
6192 6 + 19 + 2 • = . 37840 3 × (7 + 8 + 40) 6237 6 + 2 + 37 • = . 48510 4 × 85 + 10
•
• 5238 5 + 2 × (3 + 8) = . 74690 7 × (46 + 9) + 0 • •
(5 × 3 + 1) × 2 5312 = . 48970 4×8×9+7+0 •
5324 5×3×2+4 • = . 79860 7×9×8+6+0
•
•
5346 5+3×4×6 = . 21870 21 × (8 + 7) + 0 5372 5 + 37 × 2 = . 48960 48 × (9 + 6) + 0
•
•
•
6237 6×2+3×7 • = . 54810 5 × (48 + 10)
5238 5×2+3+8 • = . 17460 1 × 7 × (4 + 6) + 0
•
6+9+7+2 6972 = . • 81340 8 × (1 + 34) + 0
•
6318 6 + 3 + 18 = . 25740 2 × 5 × (7 + 4) + 0 6345 6 × (3 + 4) + 5 = . 17280 (1 + 7) × 2 × 8 + 0 6×3+4+5 6345 = . 19270 1 + 9 + 2 + 70 6419 6+4×1×9 = . 82530 8 + 2 + 530 6435 (6 + 4 + 3) × 5 = . 17820 178 + 2 + 0
6435 6×4+3×5 • = . 28710 2 × 87 × 1 + 0 6524 6 + 5 + 24 = . • 13980 1×3+9×8+0
•
•
7125 (7 + 1 + 2) × 5 = . 64980 6 × (4 + 9 × 8) + 0 7128 7 + 1 + 28 = . 65340 6 × (5 × 3 + 40)
7298 72 + 9 + 8 = . 14350 (1 + 4) × 35 + 0 7326 7 + (3 + 2) × 6 = . 84150 (84 + 1) × 5 + 0
5376 5×3+7+6 = . 12480 1+2×4×8+0
•
6534 6+5×3×4 = . 17820 178 + 2 + 0
•
•
5396 5+3×9+6 = . 12780 1 + 2 + 7 + 80
•
6591 6×5+9×1 = . 28730 2+8×7×3+0
•
7395 7 × 3 + 95 = . 14280 14 × 2 × 8 + 0
•
8192 8 × (1 + 9 + 2) = . 53760 5×3×7×6+0
•
5719 5+7+1×9 = . 32680 (3 + 2 × 6) × 8 + 0
•
6724 6 + 72 + 4 = . 38950 3 + 8 × (9 + 50)
•
7395 7 + 39 + 5 = . 21460 2 × (14 + 60)
•
8239 8 × 2 + 39 = . 67410 6 × (74 + 1) + 0
•
5×7+3×4 5734 = . 21960 (21 + 9) × 6 + 0
•
(6 + 7) × (3 + 4) 6734 = . 12580 1 × 2 × (5 + 80)
•
7395 7 × 3 + 95 = . 41820 41 × 8 × 2 + 0
•
8×2×4+9 8249 = . 13560 (1 + 3) × 5 × 6 + 0
•
7392 73 + 92 = . 14560 (1 + 4) × (5 + 60)
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
8274 8 × (2 + 7) × 4 = . 96530 96 × (5 + 30)
•
9164 9×1×6+4 = . 85320 8 + 532 + 0
•
9741 97 + 4 + 1 = . 28650 (2 + 8) × 6 × 5 + 0
•
12654 1+2+6×5+4 = . 83790 8 + 3 × 79 + 0
•
8295 8 + (2 + 9) × 5 = . 34760 3 × (4 × 7 + 60)
•
9237 9 × (2 + 3 + 7) = . 61580 6 × 15 × 8 + 0
•
9765 9×7+6×5 = . 21840 21 × 8 + 40
•
12749 (1 + 2 × 7 + 4) × 9 = . 53680 5×3×6×8+0
9315 9 + 3 × 15 • = . 42780 (4 + 27) × 8 + 0
•
9785 9×7+8+5 = . 12360 12 × 3 + 60
•
12 × (7 + 6 × 5) 12765 = . 83490 8 × (3 + 4 × 90)
9328 9 × (3 + 2) + 8 • = . 45760 4 × 5 × (7 + 6) + 0
•
135 1+3+5 = . 846720 84 × 672 + 0
•
12789 12 + 7 × 8 + 9 = . 36540 (36 × 5) + 40
1×4+8 148 = . 693750 6 × 9375 + 0
•
9367 9 + 36 × 7 = . 58140 5 × 81 × 4 + 0
•
12834 1 + 2 × (8 + 3 + 4) = . 57960 5 × (7 + 9) + 60
216 2+1+6 = . 735840 73 × 5 × 84 + 0
•
9415 9×4+1+5 = . 72630 (7 + 2) × (6 + 30)
•
12876 1 + 2 × (8 + 7) + 6 = . 53940 5 × (3 × 9 + 4) + 0
•
261 2+6+1 = . 958740 95 × 87 × 4 + 0
•
12936 1 + 2 + 9 × (3 + 6) = . 85470 8 + 547 + 0
315 3+1×5 • = . 846720 8 × 4 × 672 + 0
•
12987 129 + 8 × 7 = . 45630 4 × 5 + 630
•
132 + (5 + 4) 13254 = . 78960 7 × 8 × (9 + 6) + 0
•
13572 1 + 3 × (5 + 7) + 2 = . 46980 46 + 9 + 80
•
13 × (6 + 2 × 4) 13624 = . 75980 7 × 5 + 980
•
13728 (1 + 3) × (7 × 2 + 8) = . 56940 5 × (69 + 4) + 0
•
13746 13 + (7 + 4) × 6 = . 29580 2 × 9 × 5 + 80
•
13824 1 × 3 × 8 × 24 = . 57960 5 × 7 × (9 + 60)
•
13824 1 × 3 × 8 × 24 = . 95760 95 × 7 × 6 + 0
921 9+2+1 = . 865740 8 × 6 × 5 × (7 + 40)
•
14637 (1 + 4 × 6) × 3 + 7 = . 89250 (8 + 92) × 5 + 0
•
9×2+4 924 = . 715680 71 × 5 × 6 × 8 + 0
•
14985 1 + 4 × (9 + 8) + 5 = . 26730 (2 + 6 × 7) × 3 + 0
9684 9+6+8+4 = . 75320 7 × (5 × 3 × 2 + 0)
•
945 9 + 45 = . 127680 12 × 76 × 8 + 0
•
15624 (1 + 5 × 6) × (2 + 4) = . 79380 7 + 938 + 0
8352 8 × 3 + 52 • = . 41760 (4 + 1) × 76 + 0
•
•
•
•
•
•
•
8 + (3 + 7) × 2 8372 = . 14950 1+4+9×5+0 8415 8 + (4 + 1) × 5 = . 23970 2 × (3 + 9) + 70 8415 8+4+1×5 = . 26730 2 × (6 + 7 × 3) + 0
•
•
8+4+1+5 8415 = . 72930 7 × 2 × 9 + 30
•
8432 8 + 4 × 32 = . 69750 (6 + 9) × 75 + 0
•
8437 8 + 4 × 37 = . 51920 5 × 192 + 0
•
8514 8 + 5 × (1 + 4) = . 29670 2×9×6+7+0
•
9416 9 × (4 + 1 + 6) = . 38520 385 + 20 9417 9×4×1+7 = . 32850 3 × (2 + 8) × 5 + 0 9425 9+4+2+5 = . 86710 8 × (6 + 7 + 10)
3+1×8 318 = . • 457920 4 × 5 × 792 + 0 372 3+7+2 • = . 954180 95 × 4 × (1 + 80)
9471 9+4+7+1 = . 58630 5 × (8 + 6 × 3) + 0 •
•
•
8 + (6 + 1) × 3 8613 = . 47520 4 × 7 × 5 + 20
•
8625 8 + 6 × (2 + 5) = . 17940 1 + 7 × 9 + 40
•
8634 8 + 6 + 34 • = . 71950 (71 + 9) × 5 + 0 8692 8 + 6 + 92 = . • 14350 (1 + 4) × 35 + 0
•
8712 8+7×1×2 = . 53460 5 × (3 + 4 × 6) + 0
9472 (9 + 4 × 7) × 2 = . 18560 1 × 85 + 60 9+6×4×8 9648 = . 17520 1 + 7 × 52 + 0
9648 9 + 6 + 48 • = . 37520 3 × 75 + 20
•
•
9673 (9 + 6 × 7) × 3 = . 28450 (2 + 8) × 45 + 0 9675 9 + 6 + 75 = . 41280 4 × 12 × 8 + 0
•
•
•
•
429 4+2×9 = . 156780 15 × 67 × 8 + 0 463 4 × (6 + 3) = . 152790 15 × (2 + 790) 732 7+3×2 = . 541680 5 × (4 × (1 + 6 × 80)) 846 8 + 46 = . 293750 2 × 9375 + 0
8742 (8 + 7) × 4 + 2 = . 91650 (9 + 1) × 65 + 0
•
•
8925 8 + 92 + 5 = . 17340 17 × 3 × 4 + 0
•
9724 9 × (7 + 2 + 4) = . 63580 (6 + 3) × (5 + 80)
•
12384 (12 + 3 × 8) × 4 = . 96750 (9 + 6) × 75 + 0
•
16245 1×6+2×4+5 = . 83790 8 + (3 + 7) × 9 + 0
•
9152 9 × (1 + 5 × 2) = . 87360 (8 + 7) × (3 + 60)
•
9734 9+7×3×4 = . 12560 12 × 5 + 60
•
12389 12 + 3 + 89 = . 47650 (4 + 76) × 5 + 0
•
16375 1 × 63 + 7 + 5 = . 24890 2 × (48 + 9 + 0)
•
9 + 15 × 4 9154 = . 27860 (27 + 8) × 6 + 0
•
97 + 35 9735 = . 86140 8 × (6 + 140)
•
12495 12 × (4 + 9 × 5) = . 38760 3 × 8 × 76 + 0
•
16385 1×6×3+8×5 = . 49720 4 × 9 + 7 × 20
•
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
16835 (1 + 6) × 8 + 35 = . 24790 2 × (4 + 7 × 9) + 0
•
19764 1 + 9 × (7 + 6) + 4 = . 58320 5 × 8 + 320
•
29367 2 × (9 + 36 + 7) = . 45180 4×5×1×8+0
•
41293 4 × 1 × 29 + 3 = . 86750 (8 + 6 × 7) × 5 + 0
•
16974 (1 + 6 + 9 + 7) × 4 = . 25830 2 × (5 × 8 + 30)
•
19764 19 + 7 + 6 + 4 = . 82350 (8 + 2) × 3 × 5 + 0
•
29376 2×9+3+7+6 = . 51840 5 × 1 × (8 + 4) + 0
•
41375 (4 + 1 + 3 + 7) × 5 = . 62890 6 × (2 + 8 + 9) + 0
29736 2 + (9 + 7) × 3 + 6 • = . 58410 5 × (8 + 4 + 10)
•
41392 4 × 1 × 39 × 2 = . 87560 8 × 75 + 60
•
41538 4 + 15 + 3 × 8 = . 62790 (6 + 2) × 7 + 9 + 0
•
4×1×7×9+6 41796 = . 58320 5 × 8 + 320
•
41923 4 × 1 × 92 + 3 = . 58760 5 × 8 × (7 + 6) + 0
•
4+2+5×9+8 42598 = . 61370 6 × 13 + 7 + 0
•
42816 4 + 2 × (8 + 1 × 6) = . 73590 7+3+5×9+0
17328 (1 + 7) × (3 + 2 × 8) • = . 69540 6 × 95 + 40
•
17342 1 + (7 + 3 + 4) × 2 = . 95680 (9 + 5 + 6) × 8 + 0
19845 1 × 9 + 84 + 5 • = . 26730 (2 + 6 × 7) × 3 + 0
•
17458 1 × 74 + 5 + 8 = . 69230 69 × (2 + 3) + 0
•
•
17458 1 × 74 + 5 + 8 = . 96320 96 × (3 + 2) + 0
•
1 + 75 + 23 17523 = . 84960 8 × 4 × (9 + 6) + 0
•
19845 1 × 9 × (8 + 4 × 5) = . 32760 32 × (7 + 6) + 0
•
29853 29 + 8 × (5 + 3) = . 67410 6 × 7 × (4 + 1) + 0 31248 31 + 248 = . 95760 9 × (5 × 7 + 60)
21536 2 + 1 + 5 × (3 + 6) = . 87490 (8 + 7) × (4 + 9) + 0
•
•
21546 (2 + 1 + 5) × 4 + 6 = . 79380 7 × (9 + 3 + 8) + 0
•
31248 (3 + 1) × (2 + 4 × 8) = . 97650 (9 + 76) × 5 + 0
•
21546 2 + 1 + 54 + 6 = . 83790 8 + 3 × 79 + 0
•
3 + 19 × 2 × 8 31928 = . 45760 4 × 5 + 7 × 60
1+7+6×2+5 17625 = . • 39480 (3 + 9) × 4 + 8 + 0
(2 + 1 × 6) × 4 + 5 21645 = . • 39780 3+9+7×8+0
•
3 × (2 + 1 + 5 × 4) 32154 = . 76890 7 + 68 + 90
•
42873 (4 + 2 + 8 × 7) × 3 = . 69150 6 × (9 + 1) × 5 + 0
17639 17 + 6 + 39 • = . 28450 (2 × 8 + 4) × 5 + 0
2+1×7×4+9 21749 = . • 83650 (8 × 3 + 6) × 5 + 0
32967 3 × (2 × (9 + 6) + 7) • = . 41580 4 × 15 + 80
•
4×5+2×7+9 45279 = . 63180 6 + 3 × 18 + 0
•
45287 4 + 5 × (28 + 7) = . 93610 9 + 361 + 0
•
45936 4+5×9+3+6 = . 71280 7 + 1 + 2 + 80
•
4 + (65 + 1) × 2 46512 = . 83790 8 + 3 × 79 + 0
•
46793 4+6×7+9×3 = . 51280 5×1×2×8+0
•
(4 + 7 + 1) × 9 + 6 47196 = . 53820 5 × (3 × 8 + 2) + 0
•
4×8+7+9×2 48792 = . 51360 51 + 3 + 6 + 0
•
49368 49 + (3 + 6) × 8 = . 57120 5 × (7 + 1 + 20)
•
52164 (5 + 2 + 16) × 4 = . 79380 7 × (9 + 3 + 8) + 0
•
52839 5 × 2 + 8 + 39 = . 74160 74 + 1 × 6 + 0
39168 (3 × 9 + 1 + 6) × 8 = . 54720 5 × (4 + 72) + 0
•
53176 (5 + 3 + 1) × 7 + 6 = . 92480 (9 + 2 + 4) × 8 + 0
•
53274 5 × 3 + 2 + 74 = . 81960 8 × (1 + 9) + 60
•
57216 5 + (7 × (2 + 1) + 6) = . 98340 (9 + 8) × 3 + 4 + 0
2315 2 + 3 + 15 = . 476890 4 + 7 × 6 × (8 + 90)
34182 3 × (4 + 1 + 8 × 2) = . 75960 7 × (5 + 9 + 6) + 0
1+7×9+8+2 17982 = . 36450 3 × (6 + 4) × 5 + 0
•
•
18256 (1 × 8 + 2) × 5 + 6 = . 37490 3 × 7 + 4 + 90
•
235 + 7 + 1 23571 = . 69840 (6 + 9) × (8 + 40)
•
•
1×8+4×6+2 18462 = . 59730 5 × (9 + 7) + 30
•
23598 2 × 3 × (5 + 9) + 8 = . 46170 4 + 6 + 170
•
3 × 5 + (8 + 6) × 2 35862 = . 91740 (9 + 1) × (7 + 4) + 0
•
18492 (18 + 49) × 2 = . 36570 3 × 65 + 70
•
23715 2 + (3 + 7) × (1 + 5) = . 48960 4 × (8 + 9) + 60
•
36921 36 × (9 + 2) + 1 = . 54870 5 × (48 + 70)
•
1×8+5×6+4 18564 = . 72930 72 + 93 + 0
•
26145 2 × (61 × 4 + 5) = . 79380 7×9×3×8+0
•
3 + (7 + 5 + 8) × 2 37582 = . 96140 9 + 61 + 40
•
18792 1 + 8 × (7 + 9 + 2) • = . 35640 35 + 6 × 40
•
18942 189 + 42 = . 35670 3 × 5 + 6 × 70
26784 2 × (6 + 7 × 8) × 4 • = . 35910 35 × (9 + 10)
•
•
35192 (3 + 5 + 1) × 9 + 2 = . 48760 48 + 7 + 60
38715 3 + 8 + 71 + 5 • = . 69420 6 × (9 + 4) × 2 + 0
26871 2 × (68 × 7 + 1) = . 35490 35 × 4 × 9 + 0
•
27495 2 + (7 + 49) × 5 = . 38610 386 + 10
•
38724 3 + 8 + 7 + 24 = . 69150 (6 + 9 × 1) × 5 + 0
18954 (1 + 8 × 9 + 5) × 4 = . 26730 2 + 6 × 73 + 0
•
•
19467 1 × 9 × 4 + 67 = . 28350 (2 + 8) × 3 × 5 + 0
•
27531 (2 × 7 + 5) × 3 × 1 = . 86940 86 + 94 + 0
•
39852 3+9×8+5+2 = . 46170 4 × 6 + 1 + 70
•
19584 1 + 95 + 8 × 4 = . 67320 6 × 73 + 2 + 0
•
27936 2 × (79 + 3 × 6) = . 51840 5 × 18 × 4 + 0
•
39861 (3 + 9) × 8 + 6 + 1 = . 42570 4 × 2 × 5 + 70
•
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
58941 5 + ((8 + 9) × 4 + 1) = . 63720 6 + 37 × 2 + 0
•
2781 2+7+8+1 = . 435690 4 × (3 × 5 + 690)
•
3978 3+9+7×8 = . 146250 1 × 4 × 625 + 0
•
6318 6 + (3 + 18) = . 452790 (4 × 52 + 7) × 9 + 0
•
64125 64 + 1 + 2 × 5 = . 83790 8 + (3 + 7) × 9 + 0
•
2871 2 + 8 × (7 + 1) = . 354960 3 × 5 × (4 + 9 × 60)
•
4158 (4 + 1) × 5 + 8 = . 372960 (3 + 7) × 296 + 0
•
6327 6+3×2+7 = . 149850 1 + 49 + 8 × 50
•
65 + 1 × 8 × 4 65184 = . 73920 (7 + 3) × (9 + 2) + 0
•
3149 31 + 4 × 9 = . 285760 2 × 8 × 5 × 76 + 0
•
6327 6+3×2+7 4235 4+2+3×5 = . • 159840 = (1 + 5 + 9) × 8 × 4 + 0 . 698170 6 × (9 + 8 × (1 + 70))
•
65372 (6 + 53) × (7 + 2) = . 94180 9 × (4 + 1 + 80)
•
3179 3 + 1 × 79 = . 826540 82 × 65 × 4 + 0
•
4275 4×2+7+5 = . 186390 1 × 863 + 9 + 0
•
6327 6 × (3 + 2) + 7 = . 519840 5 × 19 × 8 × 4 + 0
•
68931 6 + (8 + 9) × (3 + 1) = . 74520 7 × 4 + 52 + 0
•
3186 3×1×8+6 = . 249570 24 × 95 + 70
•
4378 4+3+7+8 = . 296510 296 × 5 + 10
•
6 × (3 + 4) + 5 6345 = . 127980 12 × (7 + 9 × 8) + 0
•
69148 6 + (9 + 1 + 4) × 8 = . 73250 73 + 2 + 50
•
3×4×6+5 3465 = . 287910 28 + 7 × 910
•
4392 43 + 9 × 2 = . 158760 15 × (87 + 60)
•
6345 (6 + 3) × 4 × 5 = . 279180 2 + 7918 + 0
•
71495 71 + 49 × 5 = . 83260 8 + 3 × 2 × 60
•
3478 3×4+7+8 = . 591260 5 × (912 + 6 + 0)
4623 46 × 2 × 3 • = . 158790 15 × 8 × 79 + 0
•
6345 6 + 3 × 45 = . 287910 28 + 7 × 910
•
7 + (2 + 6) × 4 × 5 72645 = . 98310 9 × 8 × 3 + 10
•
3582 3+5+8×2 = . 967140 9 × (6 + 714 + 0)
•
6432 64 × 3 × 2 = . 158790 15 × 8 × 79 + 0
•
73152 7 × (31 + 5) + 2 = . 96480 9 + 6 + 4 × 80
•
3627 3 × (6 + 2) + 7 = . 189540 (1 + 8) × 9 × 5 × 4 + 0
•
6478 64 + 7 + 8 = . 152930 1 × 5 + 2 × 930
•
75816 (7 + 5 × (8 + 1)) × 6 = . 92340 (92 + 3) × 4 + 0
•
36 + 45 3645 = . 287910 28 + 7 × 910
•
6492 (6 × 4 + 9) × 2 = . 178530 1785 + 30
•
6+7+5×4 6754 = . 319280 (3 + 192) × 8 + 0
•
6825 6 × (8 + 2) + 5 = . 419370 4 + 19 × 3 × 70
•
6+8+4×2 6842 = . 519370 5 × (1 + 9 × 37) + 0
•
6915 6+9+1+5 = . 387240 3 × 8 × (7 + 2 + 40)
•
7215 7 + 2 + 15 = . 836940 8 × 3 + 69 × 40
•
79135 7 × (9 + 1 × 3 + 5) = . 82460 8 × 2 × 4 + 60
•
4 × (6 + 2) + 3 4623 = . • 785910 7 × 85 × (9 + 1 + 0) 4671 4+6+7+1 • = . 389250 3 × (8 + 92) × 5 + 0
•
3645 3 + 6 + 45 = . 289170 28 × 9 × 17 + 0 •
•
81567 8×1×5+6+7 = . 92340 (9 + 2 × 3) × 4 + 0
•
3645 36 × (4 + 5) = . 712890 712 × 89 + 0 •
81765 8+1+7+6×5 • = . 92430 9 × 2 + 4 + 30 1638 1×6×3+8 • = . 459270 45 × (92 + 70)
•
1692 1×6+9+2 = . 837540 8375 + 40
1836 18 × 3 + 6 = . • 792540 7 × 925 × 4 + 0
•
3654 3+6+5×4 = . 197820 1 + 9 + 78 × 20 •
•
•
3×6+5+8 3658 = . 149270 1 + 4 + 9 × 2 × 70 3×6+7+5 3675 = . 298410 29 × 84 × 1 + 0
3678 3×6+7+8 • = . 514920 5 × 1 × (4 + 920)
•
4725 4×7+2+5 = . 136890 13 × (6 + 8 × 9) + 0 4972 4+9+7+2 = . 163850 (1 + 6 × 3 × 8) × 5 + 0 (5 + 1 × 2) × 4 5124 = . 376980 3 × 76 × 9 + 8 + 0 5184 5+1+8+4 = . 372960 37 × (29 + 6 + 0) 5273 5×2+7+3 = . 896410 8 × (9 + 6 + 410)
•
5618 5+6×1×8 = . 473290 47 × (3 + 2 + 90)
•
7245 7×2+4+5 = . 839160 8 × (3 × 91 + 60)
•
2316 2+3+1+6 = . 584790 5 × 84 × 7 + 90
•
3726 (3 + 7 + 2) × 6 = . 194580 1 × 94 × 5 × 8 + 0
•
5814 5+8+1×4 = . 237690 23 + 7 × (6 + 90)
•
7315 7 × (3 + 1) + 5 = . 968240 (96 + 8) × (2 + 40)
•
2385 2×3×8+5 = . 697410 6 × 9 × 7 × 41 + 0
•
3+7+4+9 3749 = . 182560 (18 + 2) × 56 + 0
•
5814 5+8+1×4 = . 379620 37 × (9 + 6) × 2 + 0
•
7326 7+3+2×6 = . 149850 1 + 49 + 8 × 50
•
2514 2×5+1+4 = . 389670 3 × (8 × 96 + 7 + 0)
•
5872 (5 + 8 + 7) × 2 3912 3+9+1×2 = . • 139460 = 1 + 3 + 946 + 0 . 547680 5 × 4 × 7 × (6 + 8 + 0)
•
7 × 3 × (9 + 2) 7392 = . 156480 15 × (6 + 4 × 80)
•
2×7+6+4 2764 = . 158930 15 × (89 + 3 + 0)
•
3918 3+9+1×8 = . 254670 25 × 4 × (6 + 7) + 0
•
7865 7 × 8 + 65 = . 241930 2 + 4 × 1 × 930
•
6×1×7×2 6172 = . 354890 35 × (48 + 90)
Inder J. Taneja
RGMIA Research Report Collection, 19(2016), pp.1-80, http://rgmia.org/v19.php
•
7935 7+9+3+5 = . 682410 6 × 8 × (2 + 41) + 0
•
8 + 62 + 5 8625 = . 149730 14 × (9 × 7 + 30)
•
9182 9+1×8+2 = . 734560 73 × 4 × 5 + 60
•
7938 7+9×3+8 = . 215460 21 × 54 + 6 + 0
•
8625 8 + 6 × (2 + 5) = . 197340 1 + 9 × (7 + 3 × 40)
•
9315 93 + 15 = . 467820 4 × 678 × 2 + 0
(8 + 3 × 1) × 6 8316 = . 459270 45 × (9 + 2 + 70)
•
8721 8+7+2×1 = . 569430 5 × 6 × 9 × 4 + 30
•
•
•
8325 8 + 32 + 5 = . 176490 (17 × 6 + 4) × 9 + 0
8541 8 × (5 + 4) + 1 • = . 279630 2 + 796 × 3 + 0 •
8613 86 + 13 = . 247950 (2 + 4 × 7) × 95 + 0
•
8742 87 + 4 + 2 = . 153690 15 + 3 × 6 × 90
9135 9 + (1 + 3) × 5 • = . 786240 78 × (6 + 2) × 4 + 0
•
•
9435 9 × (4 + 3) + 5 = . 168720 16 × (8 × 7 + 20)
•
9486 9 + 48 + 6 = . 521730 5 + 2 × 1730
•
9+5+3×6 9536 = . 172840 1 × 72 × 8 + 4 + 0
•
9637 9 + 6 × (3 + 7) = . 582410 5 × (824 + 10)
•
9675 9 + 6 + 75 = . 123840 (1 + 23) × (8 + 40)
9+3×6+7 9367 = . 154280 1 × 5 × 4 × 28 + 0
•
9367 9+3×6+7 = . 451820 4 × 5 × 1 × 82 + 0
•
9432 9 + 43 × 2 = . 518760 5 + 1 × 87 × 60
94 + 3 + 5 9435 9 + (1 + 3) × 5 9135 = . • = . 128760 12 × (8 × 7 + 60) 846720 8×4×6×7×2+0
Acknowledgement The author is thankful to T.J. Eckman, Georgia, USA (email:
[email protected]) in programming the script to develop these representations.
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