Multi-Digits Magic Squares - RGMIA

RGMIA Collection, 18(2015), pp.1-22, http://rgmia.org/v18.php

Received 29/10/15

Multi-Digits Magic Squares Inder J. Taneja1) Abstract. In this work, we shall construct magic squares written in multi-digits formats. In some situations, different digits are considered according to order of magic squares. Magic squares related to combination with repetitions are also considered. In each case study is extended to palindromic magic squares.

1. Introduction Magic squares are generally constructed using sequential or continued numbers such as 1,2,…n2. Here in this work we shall write magic squares using multi-digits in a symmetrical form or equally distributed. By symmetrical form or equally distributed we understand that for each magic square, the number of digits in each cell is of same order, for example, 3-digits, 112, 121,133, etc; 4-digits, 1441, 2331, 1115, etc.; 5-digits, 25532, 12221, 23334, etc. This we have considered in three forms: (i)

Combinations without Repetitions – These are understood as magic squares formed by combinations of digits without repetitions, where in each cell there is no repetition, such as 1243, 1423, 1234, etc. Number of digits considered in each cell are of same order as of magic square. For simplicity, we shall write as different digits magic squares. In case of order 3, we know that there are only 3!=6 different digits having the three numbers 1, 2 and 3, such as, 123, 132, 213, 231, 312 and 321. For further orders, we have much more possibilities, i.e., for order 4: 4!=24; for order 5: 5!=120; for order 6: 6!=720, etc. We have given different digits magic squares for the orders 4 to 10.

(ii)

Combinations with repetitions – These kind of magic square are considered writing combinations with repetitions, such as, 112, 121, 122, etc. See the table below: Possibilities 33=27 43=64=82 34=81=92 44=256=162 54=625=252 ….

(iii)

Combinations 1,2,3 – 3 by 3 1,2,3,4 – 3 by 3 1,2,3 – 4 by 4 1,2,3,4 – 4 by 4 1,2,3,4,5 – 4 by 4 ….

Examples 123, 113, 133, etc 123, 124, 113, etc. 1123, 1223,1233, etc 1234, 1123,1114, etc 12345, 11234,11145, etc …. Table 1

Magic square Order 8 Order 9 Order 16 Order 25 ….

Palindromic – Here we construct palindromic magic squares. If we want to write 3-digits palindromes using only 1 and 2 we have exactly four (22) palindromes, i.e., 111, 121, 212 and 222. The same is true for 4-digits. The table below give exact quantity in each case:

______________________________________________________________________________________________________________________________________________________________ 1

Formerly, Professor of Mathematics, Federal University of Santa Catarina, 88040-400 Florianópolis, SC, Brazil. Email: [email protected]

1


3-digits palindromes using numbers 1, 2 1, 2, 3 1, 2, 3, 4 1, 2, 3, 4, 5 1, 2, 3, 4, 5, 6 1, 2, 3, 4, 5, 6, 7 1, 2, 3, 4, 5, 6, 7,8 1, 2, 3, 4, 5, 6, 7, 8, 9 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Table 2

Total Palindromes 4 = 22 9 = 32 16 = 42 25 = 52 36 = 62 49 = 72 64 = 82 81 = 92 90

This idea is to construct palindromic magic squares in each case. Table 2 relates only 3digits palindromes. See the table below giving exact values with higher order palindromes resulting in interesting magic squares. Possibilities 7-digits (1,2) 5-digits (1,2,3,4) 7-digits (1,2,3) 7-digits (1,2,3,4) 7-digits (1,2,3,4,5) 9-digits (1,2,3,4) 7-digits (1,2,3,4,5,6) 7-digits (1,2,3,4,5,6,7) 7-digits (1,2,3,4,5,6,7,8) 7-digits (1,2,3,4,5,6,7,8,9) ….

Total Palindromes 16=42 64=82 81=92 256=162 625=252 1024=322 1296=362 2401=492 4096=642 6561=812 …. Table 3

Magic squares obtained under above three situations are considereds “multi-digits magic squares”. In this work, we shall construct magic squares of orders 3-10, 16 and 25 using all three possibilities given in section 1, except the orders 16 and 25, where we have obtained results for the situations given in (ii) and (iii).

2. Magic Squares Order 3 For order 3, we have only 3!=6 different digits having the three numbers 1, 2 and 3 given as: 123, 132, 213, 231, 312 and 321. We observe that in 6 digits adding 111, 222 and 333, still we don’t have magic square. On the other side if we allow repetitions, we have 33  27 , given by 111 112 113 121 122 123 131 132 133 211 212 213 221 222 223 231 232 233 311 312 313 321 322 323 331 332 333

Table 4 Each row of Table 4 gives us a magic square of order 3 given in result below. Result 1. Magic squares of order 3 based on values of each row of above table 4 are given by ______________________________________________________________________________________________________________________________________________________________ 1


2


121 113 132 366

133 122 111 366

112 131 123 366

366 366 366 366 366

221 213 232 666

233 222 211 666

666 666 666 666 666

212 231 223 666

321 313 332 966

333 322 311 966

312 331 323 966

966 966 966 966 966

We have exactly 9 palindromes of 3-digits using numbers 1, 2 and 3, resulting in a magic square. See result below. Result 2. 3-digits palindromic magic square of order 3 with numbers 1, 2 and 3 is given by 212 131 323 666

333 222 111 666

666 666 666 666 666

121 313 232 666

3. Magic Squares Order 4 This section deals with magic squares of order 4 in two diffierent forms. One on different digits and second on palindromic numbers with 3 and 4-digits.

3.1.

Different digits

We have 4! =24 combinations of different digits having only four numbers, 1, 2, 3 and 4. These combinations are as follows: 1234 1243 1324 1342 1412 1421 2134 2143 2314 2341 2413 2431 3124 3142 3214 3241 3412 3421 4123 4132 4213 4231 4312 4321

Out of 24, below is a magic square of order 4 using 16 different values Result 3. Magic square of order 4 with different digits [2] considering 4 by 4 out of 1,2,3 and 4: 11110 11110 11110 11110 11110 11110 11110 11110 11110 11110

1234 2413 3142 4321

4132 2341 3214 1423

4312 2143 3412 1243

1432 4213 1342 4123

Result 4. Magic square of order 4 with different digits considering 3 by 3 out of 1,2,3 and 4: 1110 1110 1110 1110 1110 1110 1110 1110 1110 1110

123 241 314 432

3.2.

413 234 321 142

431 214 341 124

143 421 134 412

1110 1110 1110 1110 1110 1110 1110 1110 1110 1110

234 413 142 321

132 341 214 423

312 143 412 243

432 213 342 123

Palindromic

We can write 3 and 4-digits pan diagonal palindromic magic square of order 4 using four digits. See the result below. ______________________________________________________________________________________________________________________________________________________________ 1


3


Result 5. 3 and 4-digits palindromic pan diagonal magic squares of order 4 with numbers 1, 2, 3 and 4 are respectively given by 232 1110 121 1110 444 1110 313

1110 1110 1110 1110 343 111 424 1110 414 242 333 1110

131 222

323 434

212 1110 141 1110

2332 11110 1221 11110 4444 11110 3113

1110 1110 1110 1110 1110

11110 11110 11110 11110 3443 1111 4224 11110 4114 2442 3333 11110

1331 3223 2112 11110 2222 4334 1441 11110

11110 11110 11110 11110 11110

Analysing 7-digits palindromes, we can make exacly 16 palindormes having the numbers 1 and 2. This gives us a pan diagonal magic square of order 4. Result 6. 7-digits palindromic pan diagonal magic squares of order 4 just with numbers 1 and 2 is given by 6666666

6666666

6666666

1221221 6666666 1112111 6666666 2222222 6666666 2111112

2122212 2211122 1121211 1212121

1111111 1222221 2112112 2221222

2212122 2121212 1211121 1122211

6666666

6666666

6666666

6666666

6666666 6666666 6666666 6666666 6666666 6666666


4.1.

Different digits

We have total 5! =120 combinations of different digits having only 5 numbers, 1, 2, 3, 4 and 5. Below is a magic square of order 5 using 25 of 120. Result 7. Different digits magic square of order 5 using 5 different digits in each cell [3] is given by 12345 12354 34512 53142 54312 54321 45321 21453 13425 32145 12435 43215 34251 35412 41352 35421 12534 52134 51234 15342 52143 53241 24315 13452 23514 166665 166665 166665 166665 166665

166665 166665 166665 166665 166665 166665 166665

There are much more possibilities of writing above type of magic square [3], only one is given to justify its existence. Result 8. Magic square of order 5 with different digits considering 4 by 4 with 1, 2, 3, 4 and 5: 16665 16665 16665 16665 16665 16665 16665 16665 16665 16665 16665 16665

2345 4321 2435 5421 2143

2354 5321 3215 2534 3241

4512 1453 4251 2134 4315

3142 3425 5412 1234 3452

4312 2145 1352 5342 3514

16665 16665 16665 16665 16665 16665 16665 16665 16665 16665 16665 16665

1234 5432 1243 3542 5214

1235 4532 4321 1253 5324

3451 2145 3425 5213 2431

5314 1342 3541 5123 1345

5431 3214 4135 1534 2351

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4


4.2.

Palindormic

We can write 3 and 5-digits pan diagonal palindromic magic square of order 5 using five digits 1, 2, 3, 4 and 5. Result 9. 3 and 5-digits palindromic pan diagonal magic squares of order 5 with 5 numbers 1, 2, 3, 4 and 5 are respectively given by

1665 1665 1665 1665

1665 1665 1665 1665 1665 222 333 444 555 1665 545 151 212 323 1665 313 424 535 141 1665 131 242 353 414 1665 454 515 121 232 1665 1665 1665 1665 1665 1665 1665

111 434 252 525 343

166665 166665 166665 166665

166665 166665 166665 166665 166665 22322 33333 44344 55355 166665 54345 15351 21312 32323 166665 31313 42324 53335 14341 166665 13331 24342 35353 41314 166665 45354 51315 12321 23332 166665 166665 166665 166665 166665 166665 166665

11311 43334 25352 52325 34343

In [10], we have given a 5-digits pan diagonal bimagic square of order 25, where each block of order 5 is also a magic square. In subsection 2.10, we have given pan diagonal palindromic magic square of order 25 with 7-digits palindromes using only five numbers 1, 2, 3, 4 and 5.


5.1.

Different digits

We have total 6! =720 combinations of different digits having six numbers, 1, 2, 3, 4, 5 and 6. Based on these combinations, here below is a magic square of order 6 [1] with combinations of six different digits in each cell. Result 10. Magic squares of order 6 with 6 different digits [1] in each cell using the numbers 1 to 6 are given by 2333331

532614 326145 261453 614532 145326 453261

245163 451632 516324 163245 632451 324516

361245 612453 124536 245361 453612 536124

653421 534216 342165 421653 216534 165342

124356 243561 435612 356124 561243 612435

416532 165324 653241 532416 324165 241653

2333331

2333331

2333331

2333331

2333331

2333331

2333331

2333331

612453 451632 326145 165324 243561 534216

361245 245163 532614 416532 124356 653421

536124 324516 453261 241653 612435 165342

453612 632451 145326 324165 561243 216534

245361 163245 614532 532416 356124 421653

2333331

2333331

124536 516324 261453 653241 435612 342165

2333331

2333331

2333331

2333331

2333331

2333331

2333331

2333331

2333331 2333331 2333331 2333331

2333331 2333331 2333331 2333331 2333331

Below are example of 5-digits magic square of order 6 choosing out of six numbers, i.e.,1, 2, 3, 4, 5 and 6. Result 11. Magic squares of order 6 with different digits considering 5 by 5 with 1, 2, 3, 4, 5 and 6 are given by

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5


233331

32614 26145 61453 14532 45326 53261

45163 51632 16324 63245 32451 24516

61245 12453 24536 45361 53612 36124

53421 34216 42165 21653 16534 65342

24356 43561 35612 56124 61243 12435

16532 65324 53241 32416 24165 41653

233331

233331

233331

233331

233331

233331

233331

233331

12453 51632 26145 65324 43561 34216

61245 45163 32614 16532 24356 53421

36124 24516 53261 41653 12435 65342

53612 32451 45326 24165 61243 16534

45361 63245 14532 32416 56124 21653

233331

233331

24536 16324 61453 53241 35612 42165

233331

233331

233331

233331

233331

233331

233331

233331

233331 233331 233331 233331

233331 233331 233331 233331 233331

Examples in above result are obtained from Result 10, just removing the first digit in each cell. First example of Result 11 is cyclic in first four digits based on first row. In [1] [5] there are 6orthognal Latin squares leading to Semi-magic Square. More situations of Results 10 and 11 can be seen in Amelia [1]. Also in [1], there are examplea of 3, 4-digits magic squares of order 6.

5.2.

Palindromic

We can write 3 and 6-digits palindromic magic squares of order 6 using six digits 1, 2, 3, 4, 5 and 6. See result below. Result 12. 3 and 6-digits palindromic magic squares of order 6 using numbers 1 to 6 are given by 2331

252 111 313 424 565 666

121 333 636 242 444 555

434 656 535 131 232 343

545 454 222 646 323 141

626 515 464 363 151 212

353 262 161 525 616 414

2331

2331

2331

2331

2331

2331

2331

2333331

121121 333333 636636 242242 444444 555555

434434 656656 535535 131131 232232 343343

545545 454454 222222 646646 323323 141141

626626 515515 464464 363363 151151 212212

353353 262262 161161 525525 616616 414414

2333331

2331

252252 111111 313313 424424 565565 666666

2331

2333331

2333331

2333331

2333331

2333231

2333331

2333331

2331 2331 2331 2331

2333331 2333331 2333331 2333331 2333331

Doubling the values of first example, we have second example. According to Table 2, we have exacly 1296  362 palindromes of 7-digits using only the numbers 1 to 6. This lead us a palindromic magic square of order 36. Its study shall be dealt elsewhere.

6. Magic Squares Order 7 This section deals with magic squares of order 7 in two diffierent forms. One on different digits and second on palindromic numbers with 3- and 7-digits.

6.1.

Different digits

We have total 7! =5040 combinations of different digits with seven numbers, 1, 2, 3, 4, 5, 6 and 7. Below are two examples of magic squares of order 7 with different digits in each cell. Result 13. Following two examples are of magic squares of order 7 with different digits [4] in each cell:

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6

RGMIA Collection, 18(2015), pp.1-22, http://rgmia.org/v18.php 31111108

3546271 2341567 4256317 7354612 7654213 4723561 1234567

1234657 4653712 6517234 2653471 5143276 3654127 7254631

1374526 6431275 3461752 5137264 4723651 7345126 2637514

5312476 5327416 6745321 3465217 1274365 3254671 5731642

4376152 7365214 7143562 3715246 3416725 3561742 1532467

7642513 3265471 1752346 6423751 1245637 5314267 5467123

7624513 1726453 1234576 2361547 7653241 3257614 7253164

31111108

31111108

31111108

31111108

31111108

31111108

31111108

31111108

31111108

3546271 2341567 4256317 7354612 7654213 4723561 1234567

1234657 4653712 7523164 2653471 4157326 3654127 7234651

1374526 6431275 3461752 5137264 4723651 7345126 2637514

5312476 5327416 4756321 3465217 1352476 3254671 7642531

4376152 7365214 7143562 3715246 3416725 3561742 1532467

7642513 3265471 2735416 6423751 2153476 5314267 3576214

7624513 1726453 1234576 2361547 7653241 3257614 7253164

31111108

31111108

31111108

31111108

31111108

31111108

31111108

31111108

31111108

31111108 31111108 31111108 31111108 31111108 31111108

and 31111108 31111108 31111108 31111108 31111108 31111108 31111108

There are much more possibilities [4], but we have written only 2 to show its existence. Result 14. In the above result remove 1st, 2nd and 3rd initial digit of each member, we get respectively 6, 5 and 4-digits magic squares of different digits using the numbers 1 to 7: 3111108 3111108 3111108 3111108 3111108 3111108 3111108 3111108 3111108 3111108 3111108 3111108 3111108 3111108 3111108 3111108

546271 341567 256317 354612 654213 723561 234567

234657 653712 517234 653471 143276 654127 254631

374526 431275 461752 137264 723651 345126 637514

312476 327416 745321 465217 274365 254671 731642

376152 365214 143562 715246 416725 561742 532467

311108 311108 311108 311108 311108 311108 311108 311108 311108 311108 311108 311108 311108 311108 311108 311108

46271 41567 56317 54612 54213 23561 34567

34657 53712 17234 53471 43276 54127 54631

74526 31275 61752 37264 23651 45126 37514

12476 27416 45321 65217 74365 54671 31642

76152 65214 43562 15246 16725 61742 32467

42513 65471 52346 23751 45637 14267 67123

24513 26453 34576 61547 53241 57614 53164

642513 265471 752346 423751 245637 314267 467123

624513 726453 234576 361547 653241 257614 253164

30108 31108 31108 31108 31108 31108 31108 30108 30108 31108 31108 31108 31108 31108 31108 31108

6271 1567 6317 4612 4213 3561 3567

4657 3712 7234 3471 3276 4127 4631

4526 1275 1752 7264 3651 5126 7514

2476 7416 5321 5217 4365 4671 1642

6152 5214 3562 5246 6725 1742 2467

2513 5471 2346 3751 5637 4267 7123

4513 6453 4576 1547 3241 7614 3164

Still, we can have magic squares of order 7 with 7 numbers 1 to 7 considering 3 by 3. We shall deal it in another work.

6.2.

Palindromic

Similar to other cases here also we have pan diagonal palindromic magic squares of order 7 with 3 and 7-digits, having numbers 1, 2, 3, 4, 5, 6 and 7. Result 15. 3 and 7-digits pan diagonal palindromic magic square of order 7 with numbers 1, 2, 3, 4, 5, 6 and 7 are given by

______________________________________________________________________________________________________________________________________________________________ 1


7


3108 3108 3108 3108 3108 3108

3108

3108

3108

3108

3108

3108

3108

111 656 424 262 737 575 343

222 767 535 373 141 616 454

333 171 646 414 252 727 565

444 212 757 525 363 131 676

555 323 161 636 474 242 717

666 434 272 747 515 353 121

777 545 313 151 626 464 232

3108

3108

3108

3108

3108

3108

3108

3108

3108

3108

31111108

3108

31111108

3108

31111108

3108

31111108

3108

31111108

3108

31111108

31111108

31111108

31111108

31111108

31111108

31111108

31111108

1144411 6544456 4244424 2644462 7344437 5744475 3444443

2244422 7644467 5344435 3744473 1444441 6144416 4544454

3344433 1744471 6444446 4144414 2544452 7244427 5644465

4444444 2144412 7544457 5244425 3644463 1344431 6744476

5544455 3244423 1644461 6344436 4744474 2444442 7144417

6644466 4344434 2744472 7444447 5144415 3544453 1244421

7744477 5444445 3144413 1544451 6244426 4644464 2344432

31111108

31111108

31111108

31111108

31111108

31111108

31111108

31111108

31111108

31111108 31111108 31111108 31111108 31111108 31111108

According to Table 3, we have exacly 2401  492 palindromes of 7-digits using the numbers 1 to 7. This lead us a pan diagonal bimagic square of order 49. Its study shall be dealt elsewhere.

7. Magic Squares Order 8 This section deals with magic squares of order 8 in three diffierent forms. One on different digits, second on palindromic numbers with 3, 5 and 8-digits, and third on combinations with repetitions using only the numbers 1, 2, 3 and 4.

7.1.

Different digits

We have total 8! = 40320 combinations of different digits having numbers 1 to 8. Based on these combinations, here below is a magic squares formed by different combinations of above eight numbers in each cell. Result 16. Different digits magic square of order 8 using the numbers 1 to 8 is given by 399999996

27856341 41278563 12345678 38527416 56381274 74563812 63812745 85234167

63412785 85634127 56781234 74163852 12745638 38127456 27456381 41678523

52781634 78163452 67412385 81634527 23456781 45678123 16745238 34127856

16345278 34527816 23856741 45278163 67812345 81234567 52381674 78563412

41638527 27416385 38167452 12785634 74123856 56741238 85674123 63452781

85274163 63852741 74523816 56341278 38567412 12385674 41238567 27816345

78523416 52341678 81274563 67852341 45238167 23816745 34567812 16385274

34167852 16785234 45638127 23416785 81674523 67452381 78123456 52741638

399999996

399999996

399999996

399999996

399999996

399999996

399999996

399999996

399999996

399999996

399999996 399999996 399999996 399999996 399999996 399999996 399999996

It is bimagic just in rows with sum: 24189044812655004. Still we are unable to bring pan diagonal and bimagic square of order 8 with different digits. Result 17. In the above result remove 1st, 2nd, 3rd and 4th members in the bignning of each number, we get respectively 7, 6, 5 and 4-digits magic squares of different digits using the numbers 1 to 8: 7856341 1278563 2345678 8527416 6381274 4563812 3812745 5234167

3412785 5634127 6781234 4163852 2745638 8127456 7456381 1678523

2781634 8163452 7412385 1634527 3456781 5678123 6745238 4127856

6345278 4527816 3856741 5278163 7812345 1234567 2381674 8563412

1638527 7416385 8167452 2785634 4123856 6741238 5674123 3452781

5274163 3852741 4523816 6341278 8567412 2385674 1238567 7816345

8523416 2341678 1274563 7852341 5238167 3816745 4567812 6385274

4167852 6785234 5638127 3416785 1674523 7452381 8123456 2741638

39999996

39999996

39999996

39999996

39999996

39999996

39999996

39999996

39999996 39999996 39999996 39999996 39999996 39999996 39999996 39999996 39999996 39999996

______________________________________________________________________________________________________________________________________________________________ 1


8


856341 278563 345678 527416 381274 563812 812745 234167

412785 634127 781234 163852 745638 127456 456381 678523

781634 163452 412385 634527 456781 678123 745238 127856

345278 527816 856741 278163 812345 234567 381674 563412

638527 416385 167452 785634 123856 741238 674123 452781

274163 852741 523816 341278 567412 385674 238567 816345

523416 341678 274563 852341 238167 816745 567812 385274

167852 785234 638127 416785 674523 452381 123456 741638

3999996

3999996

3999996

3999996

3999996

3999996

3999996

3999996

56341 78563 45678 27416 81274 63812 12745 34167

12785 34127 81234 63852 45638 27456 56381 78523

81634 63452 12385 34527 56781 78123 45238 27856

45278 27816 56741 78163 12345 34567 81674 63412

38527 16385 67452 85634 23856 41238 74123 52781

74163 52741 23816 41278 67412 85674 38567 16345

23416 41678 74563 52341 38167 16745 67812 85274

67852 85234 38127 16785 74523 52381 23456 41638

399996

399996

399996

399996

399996

399996

399996

399996

7.2.

399996 399996 399996 399996 399996 399996 399996 399996 399996 399996

3999996 3999996 3999996 3999996 3999996 3999996 3999996 3999996 3999996 3999996

6341 8563 5678 7416 1274 3812 2745 4167

2785 4127 1234 3852 5638 7456 6381 8523

1634 3452 2385 4527 6781 8123 5238 7856

5278 7816 6741 8163 2345 4567 1674 3412

8527 6385 7452 5634 3856 1238 4123 2781

4163 2741 3816 1278 7412 5674 8567 6345

3416 1678 4563 2341 8167 6745 7812 5274

7852 5234 8127 6785 4523 2381 3456 1638

39996

39996

39996

39996

39996

39996

39996

39996

39996 39996 39996 39996 39996 39996 39996 39996 39996 39996

Palindromic

Result 18. 3 and 8-digits pan diagonal palindromic bimagic squares of order 8 using numbers 1 to 8 are given by 3996 3996 3996 3996 3996 3996 3996

3996

3996

3996

3996

3996

3996

3996

3996

282 424 111 373 565 747 636 858

616 878 585 727 131 353 262 444

545 767 656 838 222 484 171 313

151 333 242 464 676 818 525 787

434 252 363 141 717 575 888 626

868 646 737 555 383 121 414 272

777 515 828 686 454 232 343 161

323 181 474 212 848 666 757 535

3996

3996

3996

3996

3996

3996

3996

3996

3996

3996

3996 3996 3996 3996 3996 3996 3996

and 399999996 399999996 399999996 399999996 399999996 399999996 399999996

399999996

399999996

399999996

399999996

399999996

399999996

399999996

399999996

28822882 42244224 11111111 37733773 56655665 74477447 63366336 85588558

61166116 87788778 58855885 72277227 13311331 35533553 26622662 44444444

54455445 76677667 65566556 83388338 22222222 48844884 17711771 31133113

15511551 33333333 24422442 46644664 67766776 81188118 52255225 78877887

43344334 25522552 36633663 14411441 71177117 57755775 88888888 62266226

86688668 64466446 73377337 55555555 38833883 12211221 41144114 27722772

77777777 51155115 82288228 68866886 45544554 23322332 34433443 16611661

32233223 18811881 47744774 21122112 84488448 66666666 75577557 53355335

399999996

399999996

399999996

399999996

399999996

399999996

399999996

399999996

399999996

399999996

399999996 399999996 399999996 399999996 399999996 399999996 399999996

Both the above pan diagonal magic square are bimagic with bimagic sums Sb88 : 2428644 and Sb88 : 24260075687432244 respectively. In both the example, each block of order 2x4 or 4x2 has the same sum as of magic square, but each block of order 4 is not a magic square. Both examples are compact of order 4. Below are examples of pan diagonal palindromic magic squares of order 8 where each block of order 4 is a magic square. Result 19. Pan diagonal palindromic magic squares of order 8 with numbers 1 to 8 are given by ______________________________________________________________________________________________________________________________________________________________ 1


9


3996 3996 3996 3996 3996 3996 3996

3996

3996

3996

3996

3996

3996

3996

3996

111 484 555 848 121 474 565 838

858 545 414 181 868 535 424 171

444 151 888 515 434 161 878 525

585 818 141 454 575 828 131 464

212 383 656 747 222 373 666 737

757 646 313 282 767 636 323 272

343 252 787 616 333 262 777 626

686 717 242 353 676 727 232 363

3996

3996

3996

3996

3996

3996

3996

3996

3996

3996

3996 3996 3996 3996 3996 3996 3996

and 399999996 399999996 399999996 399999996 399999996 399999996 399999996

399999996

399999996

399999996

399999996

399999996

399999996

399999996

399999996

11111111 48844884 55555555 84488448 12211221 47744774 56655665 83388338

85588558 54455445 41144114 18811881 86688668 53355335 42244224 17711771

44444444 15511551 88888888 51155115 43344334 16611661 87788778 52255225

58855885 81188118 14411441 45544554 57755775 82288228 13311331 46644664

21122112 38833883 65566556 74477447 22222222 37733773 66666666 73377337

75577557 64466446 31133113 28822882 76677667 63366336 32233223 27722772

34433443 25522552 78877887 61166116 33333333 26622662 77777777 62266226

68866886 71177117 24422442 35533553 67766776 72277227 23322332 36633663

399999996

399999996

399999996

399999996

399999996

399999996

399999996

399999996

399999996

399999996

399999996 399999996 399999996 399999996 399999996 399999996 399999996

Above magic squares are not bimagic but each block of order 4 is a magic square with same magic sum. Also, each block of order 2 has the same sum as of magic square of order 4. Moreover, both are compact of order 4. Considering 5-digits palindromes, we have exactly 64 palindromes using the four numbers 1, 2, 3 and 4. It brings following result. Result 20. 5-digits pan diagonal palindromic magic squares of order 8 only with four numbers 1, 2, 3 and 4 are given by 222220 222220 222220 222220 222220 222220 222220

222220

222220

222220

222220

222220

222220

222220

222220

14441 23232 11111 22322 32223 41414 33333 44144

33133 44344 32423 41214 11311 22122 14241 23432

31413 42224 34143 43334 13231 24442 12321 21112

12121 21312 13431 24242 34343 43134 31213 42424

23332 14141 22222 11411 41114 32323 44444 33233

44244 33433 41314 32123 22422 11211 23132 14341

42324 31113 43234 34443 24142 13331 21412 12221

21212 12421 24342 13131 43434 34243 42124 31313

222220

222220

222220

222220

222220

222220

222220

222220

222220

222220

222220

222220

222220

222220

222220

222220

222220

222220

11111 24442 32123 43434 11211 24342 32223 43334

44144 31413 23132 12421 44244 31313 23232 12321

23432 12121 44444 31113 23332 12221 44344 31213

32423 43134 11411 24142 32323 43234 11311 24242

13131 22422 34143 41414 13231 22322 34243 41314

42124 33433 21112 14441 42224 33333 21212 14341

21412 14141 42424 33133 21312 14241 42324 33233

34443 41114 13431 22122 34343 41214 13331 22222

222220

222220

222220

222220

222220

222220

222220

222220

222220

222220

222220 222220 222220 222220 222220 222220 222220

and 222220 222220 222220 222220 222220 222220 222220

222220 222220 222220 222220 222220 222220 222220

First example is bimagic with bimagic sum Sb88 : 7183377060 , where each block of order 2x4 or 4x2 are of same sum. The second one is not bimagic but each block of order 4 is a magic square with sum S44 : 111110 . Moreover, both are compact of order 4. ______________________________________________________________________________________________________________________________________________________________ 1


10


7.3.

Combinations with Repetitions

Making combinations of four numbers 1,2,3 and 4 with repetitions considering 3 by 3, we have exactly 64  82 possibilities. This gives us following result. Result 21. Pan diagonal magic squares of order 8 with 1, 2, 3 and 4 considering 3 by 3 are given by

2220 2220 2220 2220 2220 2220 2220

144 232 111 223 322 414 333 441

2220 2220 2220 2220 2220 2220 2220 2220 331 314 121 233 442 423 212 2220 443 422 213 141 334 311 124 2220

324 412 113 221 142 234

341 433 132 244 123 211

134 242 343 431 312 424

222 114 411 323 444 332

413 321 224 112 231 143

432 344 241 133 214 122

243 131 434 342 421 313

2220

2220 2220

2220 2220

2220 2220

2220 2220

2220 2220 2220 2220 2220 2220 2220 2220 2220 2220 2220

2220 2220

111 244 321 434 112 243 322 433

2220 2220 2220 2220 2220 2220 2220 2220 441 234 324 131 421 214 344 2220 314 121 431 224 334 141 411 2220

231 124 442 313 232 123

444 311 233 122 443 312

114 241 323 432 113 242

341 414 132 223 342 413

211 144 422 333 212 143

424 331 213 142 423 332

134 221 343 412 133 222

2220 2220 2220 2220

2220 2220 2220 2220 2220 2220 2220 2220 2220 2220 2220

The difference between above two examples is that the first one is bimagic ( Sb88 : 717060 ) with each block of order 2x4 or 4x2 having the same sum. The second example is not bimagic, but each block of order 4 is a magic square of order 4. Moreover, both are compact of order 4.

8. Magic Squares Order 9 This section deals with magic squares of order 9 in three diffierent forms. One on different digits, second on palindromic numbers with 3, 7 and 9-digits, and third on combinations with repetitions using only the numbers 1, 2 and 3.

8.1.

Different digits

We have total 9!=362880 combinations of different digits having numbers 1 to 9. Based on these combinations, below are magic squares formed by different combinations of above nine numbers in each cell. Result 22. Different digits bimagic square of order 9 using 9 numbers 1 to 9 in each cell is given by 4999999995

123456789 459783126 786129453 378612945 615948372 942375618 264597831 591834267 837261594

297531864 534867291 861294537 156489723 483726159 729153486 312645978 648972315 975318642

345678912 672915348 918342675 231564897 567891234 894237561 189423756 426759183 753186429

486729153 723156489 159483726 642975318 978312645 315648972 537861294 864297531 291534867

561894237 897231564 234567891 429753186 756189423 183426759 675918342 912345678 348672915

618942375 945378612 372615948 594837261 831264597 267591834 453786129 789123456 126459783

759183426 186429753 423756189 915348672 342675918 678912345 891234567 237561894 564897231

834267591 261594837 597831264 783126459 129453786 456789123 948372615 375618942 612945378

972315648 318642975 645978312 867291534 294537861 531864297 726159483 153486729 489723156

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995 4999999995 4999999995 4999999995 4999999995 4999999995 4999999995 4999999995

Above magic square is bimagic with bimagic sum Sb99 : 3376740371623259625 . Even though above magic square has the property that each block of order 3 has the same as of magic square, ______________________________________________________________________________________________________________________________________________________________ 1


11


but it is neither compact nor pan diagonal. Reorganizing the values of above example, we can write a compact and pan diagonal magic square of order 9. In this case, it is not bimagic. Result 23. Different digits pan diagonal magic square of order 9 is given by

4999999995 4999999995 4999999995 4999999995 4999999995 4999999995 4999999995 4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

183456729 615978342 759123486 291564837 534897261 867231594 372645918 426789153 948312675

918345672 261597834 675912348 729156483 153489726 486723159 837264591 342678915 594831267

291834567 426159783 867591234 372915648 615348972 948672315 183726459 534267891 759483126

729183456 342615978 486759123 837291564 261534897 594867231 918372645 153426789 675948312

672918345 834261597 348675912 483729156 726153489 159486723 591837264 915342678 267594831

567291834 783426159 234867591 648372915 972615348 315948672 459183726 891534267 126759483

456729183 978342615 123486759 564837291 897261534 231594867 645918372 789153426 312675948

345672918 597834261 912348675 156483729 489726153 723159486 264591837 678915342 831267594

834567291 159783426 591234867 915648372 348972615 672315948 726459183 267891534 483126759

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995 4999999995 4999999995 4999999995 4999999995 4999999995 4999999995 4999999995

The above magic square is pan diagonal but not bimagic. Also it is compact of order 3. Result 24. In the example given in Result 23, remove 1st, 2nd, 3rd and 4th members in the bignning of each member, we get respectively 8, 7, 6 and 5-digits magic squares of different digits using the numbers 1 to 9:

499999995 499999995 499999995 499999995 499999995 499999995 499999995 499999995

499999995

499999995

499999995

499999995

499999995

499999995

499999995

499999995

83456729 15978342 59123486 91564837 34897261 67231594 72645918 26789153 48312675

18345672 61597834 75912348 29156483 53489726 86723159 37264591 42678915 94831267

91834567 26159783 67591234 72915648 15348972 48672315 83726459 34267891 59483126

29183456 42615978 86759123 37291564 61534897 94867231 18372645 53426789 75948312

72918345 34261597 48675912 83729156 26153489 59486723 91837264 15342678 67594831

67291834 83426159 34867591 48372915 72615348 15948672 59183726 91534267 26759483

56729183 78342615 23486759 64837291 97261534 31594867 45918372 89153426 12675948

45672918 97834261 12348675 56483729 89726153 23159486 64591837 78915342 31267594

34567291 59783426 91234867 15648372 48972615 72315948 26459183 67891534 83126759

499999995

499999995

499999995

499999995

499999995

499999995

499999995

499999995

499999995

49999995 49999995 49999995 49999995 49999995 49999995 49999995 49999995

49999995

49999995

49999995

49999995

49999995

49999995

49999995

49999995

3456729 5978342 9123486 1564837 4897261 7231594 2645918 6789153 8312675

8345672 1597834 5912348 9156483 3489726 6723159 7264591 2678915 4831267

1834567 6159783 7591234 2915648 5348972 8672315 3726459 4267891 9483126

9183456 2615978 6759123 7291564 1534897 4867231 8372645 3426789 5948312

2918345 4261597 8675912 3729156 6153489 9486723 1837264 5342678 7594831

7291834 3426159 4867591 8372915 2615348 5948672 9183726 1534267 6759483

6729183 8342615 3486759 4837291 7261534 1594867 5918372 9153426 2675948

5672918 7834261 2348675 6483729 9726153 3159486 4591837 8915342 1267594

4567291 9783426 1234867 5648372 8972615 2315948 6459183 7891534 3126759

49999995

49999995

49999995

49999995

49999995

49999995

49999995

49999995

49999995

4999995 4999995 4999995 4999995 4999995 4999995 4999995 4999995

4999995

4999995

4999995

4999995

4999995

4999995

4999995

4999995

456729 978342 123486 564837 897261 231594 645918 789153 312675

345672 597834 912348 156483 489726 723159 264591 678915 831267

834567 159783 591234 915648 348972 672315 726459 267891 483126

183456 615978 759123 291564 534897 867231 372645 426789 948312

918345 261597 675912 729156 153489 486723 837264 342678 594831

291834 426159 867591 372915 615348 948672 183726 534267 759483

729183 342615 486759 837291 261534 594867 918372 153426 675948

672918 834261 348675 483729 726153 159486 591837 915342 267594

567291 783426 234867 648372 972615 315948 459183 891534 126759

4999995

4999995

4999995

4999995

4999995

4999995

4999995

4999995

4999995

499999995 499999995 499999995 499999995 499999995 499999995 499999995 499999995 499999995 499999995 499999995

49999995 49999995 49999995 49999995 49999995 49999995 49999995 49999995 49999995 49999995 49999995

4999995 4999995 4999995 4999995 4999995 4999995 4999995 4999995 4999995 4999995 4999995

and ______________________________________________________________________________________________________________________________________________________________ 1


12


499995

499995

499995

499995

499995

499995

499995

499995

56729 78342 23486 64837 97261 31594 45918 89153 12675

45672 97834 12348 56483 89726 23159 64591 78915 31267

34567 59783 91234 15648 48972 72315 26459 67891 83126

83456 15978 59123 91564 34897 67231 72645 26789 48312

18345 61597 75912 29156 53489 86723 37264 42678 94831

91834 26159 67591 72915 15348 48672 83726 34267 59483

29183 42615 86759 37291 61534 94867 18372 53426 75948

72918 34261 48675 83729 26153 59486 91837 15342 67594

67291 83426 34867 48372 72615 15948 59183 91534 26759

499995

499995

499995

499995

499995

499995

499995

499995

499995

499995 499995 499995 499995 499995 499995 499995 499995

499995 499995 499995 499995 499995 499995 499995 499995 499995 499995 499995

All the above magic squares are compact of order 3. Similary we can write four more magic squares with 8, 7, 6 and 5-digits using the Result 22 with different digits from 1 to 9. In each case the magic squares obtained are bimagic.

8.2.

Palindromic

Let us write palindromic magic squares of order 9 with 3 and 9-digits palindromes. Result 25. 3-digits palindromic bimagic and pan diagonal magic squares of order 9 are respectively given by 4995

111 464 787 393 626 949 252 575 838

292 525 848 151 474 737 313 666 989

353 676 939 212 565 888 191 424 747

484 717 161 646 999 323 535 858 272

545 898 222 434 757 171 686 919 363

636 959 373 585 818 262 444 797 121

767 181 414 929 343 696 878 232 555

828 242 595 777 131 454 969 383 616

979 333 656 868 282 515 727 141 494

4995

4995

4995

4995

4995

4995

4995

4995

4995

4995

4995

4995

4995

4995

4995

4995

4995

4995

4995

4995

4995

4995

4995

4995

4995

4995

4995

4995

4995

4995

4995

4995

4995

4995

4995

4995

111 424 737 343 656 969 272 585 898

242 555 868 171 484 797 313 626 939

373 686 999 212 525 838 141 454 767

434 717 121 666 949 353 595 878 282

565 848 252 494 777 181 636 919 323

696 979 383 535 818 222 464 747 151

727 131 414 959 363 646 888 292 575

858 262 545 787 191 474 929 333 616

989 393 676 828 232 515 757 161 444

4995

4995

4995

4995

4995

4995

4995

4995

4995

4995

4995

4995 4995 4995 4995 4995 4995 4995 4995

First magic square is bimagic with bimagic sum Sb99 : 3390285 but not pan diagonal. The second is pan diagonal but not bimagic. In both cases the sum of 9 members of each block of order 3 has the same sum as of magic square. Moreover, the second magic square is compact of order 3. Result 26. 9-digits palindromic bimagic and pan diagonal magic squares of order 9 are respectively given by 4999999995

111111111 444666444 777888777 333999333 666222666 999444999 222555222 555777555 888333888

222999222 555222555 888444888 111555111 444777444 777333777 333111333 666666666 999888999

333555333 666777666 999333999 222111222 555666555 888888888 111999111 444222444 777444777

444888444 777111777 111666111 666444666 999999999 333222333 555333555 888555888 222777222

555444555 888999888 222222222 444333444 777555777 111777111 666888666 999111999 333666333

666333666 999555999 333777333 555888555 888111888 222666222 444444444 777999777 111222111

777666777 111888111 444111444 999222999 333444333 666999666 888777888 222333222 555555555

888222888 222444222 555999555 777777777 111333111 444555444 999666999 333888333 666111666

999777999 333333333 666555666 888666888 222888222 555111555 777222777 111444111 444999444

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995 4999999995 4999999995 4999999995 4999999995 4999999995 4999999995 4999999995

and ______________________________________________________________________________________________________________________________________________________________ 1


13


4999999995 4999999995 4999999995 4999999995 4999999995 4999999995 4999999995 4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

111111111 444222444 777333777 333444333 666555666 999666999 222777222 555888555 888999888

222444222 555555555 888666888 111777111 444888444 777999777 333111333 666222666 999333999

333777333 666888666 999999999 222111222 555222555 888333888 111444111 444555444 777666777

444333444 777111777 111222111 666666666 999444999 333555333 555999555 888777888 222888222

555666555 888444888 222555222 444999444 777777777 111888111 666333666 999111999 333222333

666999666 999777999 333888333 555333555 888111888 222222222 444666444 777444777 111555111

777222777 111333111 444111444 999555999 333666333 666444666 888888888 222999222 555777555

888555888 222666222 555444555 777888777 111999111 444777444 999222999 333333333 666111666

999888999 333999333 666777666 888222888 222333222 555111555 777555777 111666111 444444444

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995 4999999995 4999999995 4999999995 4999999995 4999999995 4999999995 4999999995

First magic square is bimagic with bimagic sum Sb99 : 3517039990002961485 but not pan diagonal. The second is pan diagonal but not bimagic. In both cases the sum of 9 members of each block of order 3 has same sum as of magic square. Moreover, the second magic square is compact of order 3. According to Table 3, there are exactly 81 numbers of 7-digits palindormes with numbers 1, 2 and 3. Based on this, below are two palindromic magic square of order 9. Result 27. 7-digits pan diagonal palindromic magic squares of order 9 only with numbers 1, 2, and 3 are respectively given by 19999998

1111111 2123212 3132313 1333331 2312132 3321233 1222221 2231322 3213123

1233321 2212122 3221223 1122211 2131312 3113113 1311131 2323232 3332333

1322231 2331332 3313133 1211121 2223222 3232323 1133311 2112112 3121213

2132312 3111113 1123211 2321232 3333333 1312131 2213122 3222223 1231321

2221222 3233323 1212121 2113112 3122213 1131311 2332332 3311133 1323231

2313132 3322233 1331331 2232322 3211123 1223221 2121212 3133313 1112111

3123213 1132311 2111112 3312133 1321231 2333332 3231323 1213121 2222222

3212123 1221221 2233322 3131313 1113111 2122212 3323233 1332331 2311132

3331333 1313131 2322232 3223223 1232321 2211122 3112113 1121211 2133312

19999998

19999998

19999998

19999998

19999998

19999998

19999998

19999998

19999998

19999998

19999998

19999998 19999998 19999998 19999998 19999998 19999998 19999998 19999998

and 19999998 19999998 19999998 19999998 19999998 19999998 19999998 19999998

19999998

19999998

19999998

19999998

19999998

19999998

19999998

19999998

19999998

1111111 2112112 3113113 1321231 2322232 3323233 1231321 2232322 3233323

1221221 2222222 3223223 1131311 2132312 3133313 1311131 2312132 3313133

1331331 2332332 3333333 1211121 2212122 3213123 1121211 2122212 3123213

2113112 3111113 1112111 2323232 3321233 1322231 2233322 3231323 1232321

2223222 3221223 1222221 2133312 3131313 1132311 2313132 3311133 1312131

2333332 3331333 1332331 2213122 3211123 1212121 2123212 3121213 1122211

3112113 1113111 2111112 3322233 1323231 2321232 3232323 1233321 2231322

3222223 1223221 2221222 3132313 1133311 2131312 3312133 1313131 2311132

3332333 1333331 2331332 3212123 1213121 2211122 3122213 1123211 2121212

19999998

19999998

19999998

19999998

19999998

19999998

19999998

19999998

19999998

19999998

19999998

19999998 19999998 19999998 19999998 19999998 19999998 19999998 19999998

First magic square is bimagic with bimagic sum Sb99 : 50505077616162 . In both examples, each block of order 3 are of same sum as of magic square. The second one is not bimagic but pan diagonal. Moreover, second magic square is also compact of order 3.

______________________________________________________________________________________________________________________________________________________________ 1


14


8.3.

Combinations with Repetitions

By considering 3 numbers 1, 2 and 3 and making combinations with repetitions writing 4 by 4, we have exactly 34  81  92 possibilities. This gives two different kinds of magic squares of order 9. Result 28. Bimagic and pan diagonal magic squares of order 9 with 1, 2 and 3 considering 4 by 4 are respectively given by 19998

1111 2123 3132 1333 2312 3321 1222 2231 3213

1233 2212 3221 1122 2131 3113 1311 2323 3332

1322 2331 3313 1211 2223 3232 1133 2112 3121

2132 3111 1123 2321 3333 1312 2213 3222 1231

2221 3233 1212 2113 3122 1131 2332 3311 1323

2313 3322 1331 2232 3211 1223 2121 3133 1112

3123 1132 2111 3312 1321 2333 3231 1213 2222

3212 1221 2233 3131 1113 2122 3323 1332 2311

3331 1313 2322 3223 1232 2211 3112 1121 2133

19998

19998

19998

19998

19998

19998

19998

19998

19998

19998

19998

19998

19998

19998

19998

19998

19998

19998

19998

19998

19998

19998

19998

19998

19998

19998

19998

19998

19998

19998

19998

19998

19998

19998

19998

19998

1111 2112 3113 1321 2322 3323 1231 2232 3233

1221 2222 3223 1131 2132 3133 1311 2312 3313

1331 2332 3333 1211 2212 3213 1121 2122 3123

2113 3111 1112 2323 3321 1322 2233 3231 1232

2223 3221 1222 2133 3131 1132 2313 3311 1312

2333 3331 1332 2213 3211 1212 2123 3121 1122

3112 1113 2111 3322 1323 2321 3232 1233 2231

3222 1223 2221 3132 1133 2131 3312 1313 2311

3332 1333 2331 3212 1213 2211 3122 1123 2121

19998

19998

19998

19998

19998

19998

19998

19998

19998

19998

19998

19998 19998 19998 19998 19998 19998 19998 19998

First magic square is bimagic with bimagic sum Sb99 : 50496162 but not pan diagonal. The second is pan diagonal but not bimagic. In both the cases sum of 9 members of each block of order 3 has the same sum as of magic square. Moreover, the second magic square is compact of order 3.

9. Magic Squares Order 10 This section deals with magic squares of order 10 in two diffierent forms. One on different digits, and second on palindromic numbers with 3 and 10-digits.

9.1.

Different digits

We have total 10!=3628800 combinations of different digits having ten numbers, 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, including 0 in first place. Based on these combinations, here below is a magic square formed by different combinations of above ten numbers in each cell. Example 29. Different digits bimagic square of order 10 [4] is given by 49999999995

0437195628 1946280537 7895621043 6780532194 5371946280 2014378956 8503719462 9652804371 4268053719 3129467805

8053719462 2105378946 3719462805 5678943210 9562804371 6820537194 7984621053 1496280537 0347195628 4231056789

7894621053 6280537194 4328056719 0537194628 8956710432 5761943280 3179462805 2015378946 1403289567 9642805371

6789532104 5671943280 9462805371 1046289537 2805371946 0357194628 4238056719 3120467895 7914628053 8593710462

4628053719 9567804321 8953710462 7194628053 3219467805 1406289537 0342195678 5731946280 6870532194 2085371946

5371964280 0432159678 1046298537 2805317946 6780523194 8593701462 9657840321 4268035719 3129476805 7914682053

3210476895 4328065719 0537149628 8953701462 7194682053 9642850371 5761934280 6879523104 2085317946 1406298537

2105387946 3719426805 5671934280 9462850371 1043298567 4238065719 6820573194 7984612053 8596701432 0357149628

1946208537 7894612053 6280573194 4321065789 0437159628 3179426805 2015387946 8503791462 9652840371 5768934210

9562840371 8053791462 2104387956 3219476805 4628035719 7985612043 1496208537 0347159628 5731964280 6870523194

49999999995

49999999995

49999999995

49999999995

49999999995

49999999995

49999999995

49999999995

49999999995

49999999995

49999999995

49999999995

49999999995 49999999995 49999999995 49999999995 49999999995 49999999995 49999999995 49999999995 49999999995

Result 30. In the above result removing members in the bignning or at end of each member, we get respectively 9, 8, 7, 6 and 5-digits magic squares of different digits using the numbers 0 to 9: ______________________________________________________________________________________________________________________________________________________________ 1


15


4999999995

437195628 946280537 895621043 780532194 371946280 014378956 503719462 652804371 268053719 129467805

053719462 105378946 719462805 678943210 562804371 820537194 984621053 496280537 347195628 231056789

894621053 280537194 328056719 537194628 956710432 761943280 179462805 015378946 403289567 642805371

789532104 671943280 462805371 046289537 805371946 357194628 238056719 120467895 914628053 593710462

628053719 567804321 953710462 194628053 219467805 406289537 342195678 731946280 870532194 085371946

371964280 432159678 046298537 805317946 780523194 593701462 657840321 268035719 129476805 914682053

210476895 328065719 537149628 953701462 194682053 642850371 761934280 879523104 085317946 406298537

105387946 719426805 671934280 462850371 043298567 238065719 820573194 984612053 596701432 357149628

946208537 894612053 280573194 321065789 437159628 179426805 015387946 503791462 652840371 768934210

562840371 053791462 104387956 219476805 628035719 985612043 496208537 347159628 731964280 870523194

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995

4999999995 4999999995 4999999995 4999999995 4999999995 4999999995 4999999995 4999999995 4999999995

499999995

37195628 46280537 95621043 80532194 71946280 14378956 03719462 52804371 68053719 29467805

53719462 05378946 19462805 78943210 62804371 20537194 84621053 96280537 47195628 31056789

94621053 80537194 28056719 37194628 56710432 61943280 79462805 15378946 03289567 42805371

89532104 71943280 62805371 46289537 05371946 57194628 38056719 20467895 14628053 93710462

28053719 67804321 53710462 94628053 19467805 06289537 42195678 31946280 70532194 85371946

71964280 32159678 46298537 05317946 80523194 93701462 57840321 68035719 29476805 14682053

10476895 28065719 37149628 53701462 94682053 42850371 61934280 79523104 85317946 06298537

05387946 19426805 71934280 62850371 43298567 38065719 20573194 84612053 96701432 57149628

46208537 94612053 80573194 21065789 37159628 79426805 15387946 03791462 52840371 68934210

62840371 53791462 04387956 19476805 28035719 85612043 96208537 47159628 31964280 70523194

499999995

499999995

499999995

499999995

499999995

499999995

499999995

499999995

499999995

499999995

499999995

499999995

499999995 499999995 499999995 499999995 499999995 499999995 499999995 499999995 499999995

49999995

4371956 9462805 8956210 7805321 3719462 0143789 5037194 6528043 2680537 1294678

0537194 1053789 7194628 6789432 5628043 8205371 9846210 4962805 3471956 2310567

8946210 2805371 3280567 5371946 9567104 7619432 1794628 0153789 4032895 6428053

7895321 6719432 4628053 0462895 8053719 3571946 2380567 1204678 9146280 5937104

6280537 5678043 9537104 1946280 2194678 4062895 3421956 7319462 8705321 0853719

3719642 4321596 0462985 8053179 7805231 5937014 6578403 2680357 1294768 9146820

2104768 3280657 5371496 9537014 1946820 6428503 7619342 8795231 0853179 4062985

1053879 7194268 6719342 4628503 0432985 2380657 8205731 9846120 5967014 3571496

9462085 8946120 2805731 3210657 4371596 1794268 0153879 5037914 6528403 7689342

5628403 0537914 1043879 2194768 6280357 9856120 4962085 3471596 7319642 8705231

49999995

49999995

49999995

49999995

49999995

49999995

49999995

49999995

49999995

49999995

49999995

49999995

49999995 49999995 49999995 49999995 49999995 49999995 49999995 49999995 49999995

4999995

437195 946280 895621 780532 371946 014378 503719 652804 268053 129467

053719 105378 719462 678943 562804 820537 984621 496280 347195 231056

894621 280537 328056 537194 956710 761943 179462 015378 403289 642805

789532 671943 462805 046289 805371 357194 238056 120467 914628 593710

628053 567804 953710 194628 219467 406289 342195 731946 870532 085371

371964 432159 046298 805317 780523 593701 657840 268035 129476 914682

210476 328065 537149 953701 194682 642850 761934 879523 085317 406298

105387 719426 671934 462850 043298 238065 820573 984612 596701 357149

946208 894612 280573 321065 437159 179426 015387 503791 652840 768934

562840 053791 104387 219476 628035 985612 496208 347159 731964 870523

4999995

4999995

4999995

4999995

4999995

4999995

4999995

4999995

4999995

4999995

4999995

4999995

4999995 4999995 4999995 4999995 4999995 4999995 4999995 4999995 4999995

and 499995

43719 94628 89562 78053 37194 01437 50371 65280 26805 12946

05371 10537 71946 67894 56280 82053 98462 49628 34719 23105

89462 28053 32805 53719 95671 76194 17946 01537 40328 64280

78953 67194 46280 04628 80537 35719 23805 12046 91462 59371

62805 56780 95371 19462 21946 40628 34219 73194 87053 08537

37196 43215 04629 80531 78052 59370 65784 26803 12947 91468

21047 32806 53714 95370 19468 64285 76193 87952 08531 40629

10538 71942 67193 46285 04329 23806 82057 98461 59670 35714

94620 89461 28057 32106 43715 17942 01538 50379 65284 76893

56284 05379 10438 21947 62803 98561 49620 34715 73196 87052

499995

499995

499995

499995

499995

499995

499995

499995

499995

499995

499995

499995

499995 499995 499995 499995 499995 499995 499995 499995 499995

Still, there are magic squares of order 10 with ten numbers 0 to 9, considering 3 by 3 and 4 by 4. This shall be studied elsewhere. ______________________________________________________________________________________________________________________________________________________________ 1


16


9.2.

Palindromic

Example 31. 3-digits palindromic magic square of order 10 is given by 4995 4995 4995 4995 4995 4995 4995 4995 4995 4995 4995 4995 4995 4995 4995 4995 4995 4995 4995 4995 4995 4995

000 353 616 191 939 848 474 262 727 585

484 111 575 868 202 636 020 949 393 757

151 676 222 545 080 303 797 434 969 818

919 828 494 333 565 787 252 101 646 070

878 292 959 707 444 161 515 686 030 323

232 989 060 424 717 555 343 898 171 606

747 404 383 979 858 090 666 525 212 131

363 535 808 010 696 929 181 777 454 242

595 040 737 656 121 272 909 313 888 464

626 767 141 282 373 414 838 050 505 999

Since we have only 90 palindromes of 3-digits, other 10 numbers are used as 000, 010, 020, …, 090 to complete the magic square. Example 32. 10-digits palindromic magic square of order 10 having all the 10 digits is given by 49950004940

4990000994 4994224994 4996996994 4999449994 4993773994 4991881994 4997337994 4995115994 4992662994 4998558994

4996336994 4991111994 4999559994 4998778994 4997007994 4992992994 4994884994 4990660994 4993223994 4995445994

4998448994 4993663994 4992222994 4990990994 4991551994 4996116994 4995775994 4999009994 4994334994 4997887994

4991221994 4990880994 4995005994 4993333994 4998998994 4997667994 4992442994 4994554994 4999119994 4996776994

4999889994 4995335994 4998668994 4992002994 4994444994 4990770994 4991991994 4996226994 4997557994 4993113994

4997117994 4998008994 4994774994 4991661994 4996886994 4995555994 4999229994 4993993994 4990440994 4992332994

4993553994 4999779994 4997447994 4994114994 4990330994 4998228994 4996666994 4992882994 4995995994 4991001994

4994994994 4992552994 4991331994 4995885994 4999669994 4993443994 4998118994 4997777994 4996006994 4990220994

4992772994 4997997994 4990110994 4996556994 4995225994 4999339994 4993003994 4991441994 4998888994 4994664994

4995665994 4996446994 4993883994 4997227994 4992112994 4994004994 4990550994 4998338994 4991771994 4999999994

49950004940

49950004940

49950004940

49950004940

49950004940

49950004940

49950004940

49950004940

49950004940

49950004940

49950004940 49950004940 49950004940 49950004940 49950004940 49950004940 49950004940 49950004940 49950004940 49950004940 49950004940

We have written only one example just to have an idea, but there are much more possibilities of writing 10-digits palindromic magic square of order 10 as there 90000 palindromes of 10-digits.

10.

Magic Squares Order 16

This section deals with magic squares of order 16 in two diffierent forms. One on 7-digits palindormes and second on combinations with repetitions using only the numbers 1, 2, 3 and 4. For 5-digits palindromic magic squares of order 16 refer to Taneja [9].

10.1. Palindromic In [9], author wrote bimagic and pan diagonal magic squares of order 16 with 5-digits palindromes. From, Table 3, we observe that there are exactly 256 7-digits palindromes with the numbers 1, 2, 3 and 4. Based on this we have two different types of palindromic magic squares of order 16. One is bimagic and another is pan diagonal. In both the cases, the each block of order 4 is a magic square. See the results below. ______________________________________________________________________________________________________________________________________________________________ 1


17


Result 33. 7-digits palindromic bimagic square of order 16 having the numbers 1, 2, 3 and 4 is given by 1111111 4324234 2432342 3243423 1342431 4133314 2221222 3414143 1423241 4212124 2144412 3331333 1234321 4441444 2313132 3122213

3232323 2443442 4311134 1124211 3421243 2214122 4142414 1333331 3344433 2131312 4223224 1412141 3113113 2322232 4434344 1241421

4343434 1132311 3224223 2411142 4114114 1321231 3433343 2242422 4231324 1444441 3312133 2123212 4422244 1213121 3141413 2334332

2424242 3211123 1143411 4332334 2233322 3442443 1314131 4121214 2112112 3323233 1431341 4244424 2341432 3134313 1222221 4413144

1223221 4412144 2344432 3131313 1434341 4241424 2113112 3322233 1311131 4124214 2232322 3443443 1142411 4333334 2421242 3214123

3144413 2331332 4423244 1212121 3313133 2122212 4234324 1441441 3432343 2243422 4111114 1324231 3221223 2414142 4342434 1133311

4431344 1244421 3112113 2323232 4222224 1413141 3341433 2134312 4143414 1332331 3424243 2211122 4314134 1121211 3233323 2442442

2312132 3123213 1231321 4444444 2141412 3334333 1422241 4213124 2224222 3411143 1343431 4132314 2433342 3242423 1114111 4321234

1334331 4141414 2213122 3422243 1123211 4312134 2444442 3231323 1242421 4433344 2321232 3114113 1411141 4224224 2132312 3343433

3413143 2222222 4134314 1341431 3244423 2431342 4323234 1112111 3121213 2314132 4442444 1233321 3332333 2143412 4211124 1424241

4122214 1313131 3441443 2234322 4331334 1144411 3212123 2423242 4414144 1221221 3133313 2342432 4243424 1432341 3324233 2111112

2241422 3434343 1322231 4113114 2412142 3223223 1131311 4344434 2333332 3142413 1214121 4421244 2124212 3311133 1443441 4232324

1442441 4233324 2121212 3314133 1211121 4424244 2332332 3143413 1134311 4341434 2413142 3222223 1323231 4112114 2244422 3431343

3321233 2114112 4242424 1433341 3132313 2343432 4411144 1224221 3213123 2422242 4334334 1141411 3444443 2231322 4123214 1312131

4214124 1421241 3333333 2142412 4443444 1232321 3124213 2311132 4322234 1113111 3241423 2434342 4131314 1344431 3412143 2223222

2133312 3342433 1414141 4221224 2324232 3111113 1243421 4432344 2441442 3234323 1122211 4313134 2212122 3423243 1331331 4144414

The above bimagic square has the property that each block of order 4 is also a magic square. The magic sums are S1616  44444440 , Sb1616  143658905634120 and S44  11111110 . Sb means bimagic sum. Above magic square is not pan diagonal. Below is an example of pan diagonal palindromic magic square of order 16. Result 34. 7-digits pan diagonal palindromic magic square of order 16 having the numbers 1, 2, 3 and 4 is given by 2121212 1332331 4413144 3244423 2224222 1433341 4312134 3141413 2322232 1131311 4214124 3443443 2423242 1234321 4111114 3342433

3213123 4444444 1321231 2132312 3112113 4341434 1424241 2233322 3414143 4243424 1122211 2331332 3311133 4142414 1223221 2434342

1344431 2113112 3232323 4421244 1441441 2212122 3133313 4324234 1143411 2314132 3431343 4222224 1242421 2411142 3334333 4123214

4432344 3221223 2144412 1313131 4333334 3124213 2241422 1412141 4231324 3422243 2343432 1114111 4134314 3323233 2442442 1211121

2422242 1231321 4114114 3343433 2323232 1134311 4211124 3442443 2221222 1432341 4313134 3144413 2124212 1333331 4412144 3241423

3314133 4143414 1222221 2431342 3411143 4242424 1123211 2334332 3113113 4344434 1421241 2232322 3212123 4441444 1324231 2133312

1243421 2414142 3331333 4122214 1142411 2311132 3434343 4223224 1444441 2213122 3132313 4321234 1341431 2112112 3233323 4424244

4131314 3322233 2443442 1214121 4234324 3423243 2342432 1111111 4332334 3121213 2244422 1413141 4433344 3224223 2141412 1312131

2223222 1434341 4311134 3142413 2122212 1331331 4414144 3243423 2424242 1233321 4112114 3341433 2321232 1132311 4213124 3444443

3111113 4342434 1423241 2234322 3214123 4443444 1322231 2131312 3312133 4141414 1224221 2433342 3413143 4244424 1121211 2332332

1442441 2211122 3134313 4323234 1343431 2114112 3231323 4422244 1241421 2412142 3333333 4124214 1144411 2313132 3432343 4221224

4334334 3123213 2242422 1411141 4431344 3222223 2143412 1314131 4133314 3324233 2441442 1212121 4232324 3421243 2344432 1113111

2324232 1133311 4212124 3441443 2421242 1232321 4113114 3344433 2123212 1334331 4411144 3242423 2222222 1431341 4314134 3143413

3412143 4241424 1124211 2333332 3313133 4144414 1221221 2432342 3211123 4442444 1323231 2134312 3114113 4343434 1422241 2231322

1141411 2312132 3433343 4224224 1244421 2413142 3332333 4121214 1342431 2111112 3234323 4423244 1443441 2214122 3131313 4322234

4233324 3424243 2341432 1112111 4132314 3321233 2444442 1213121 4434344 3223223 2142412 1311131 4331334 3122213 2243422 1414141

The above bimagic square has the property that each block of order 4 is also a magic square. The magic sums are S1616  44444440 and S44  11111110 . Above magic square is pan diagonal but not bimagic.

10.2. Combinations with Repetitions By allowing repetitions in numbers 1, 2, 3 and 4, we have exactly 44  256  162 possibilities. These values gives us both bimagic and pan diagonal magic squares [9] given in the following results. Result 35. Bimagic square of order 16 formed by digits 1, 2, 3 and 4 is given by

______________________________________________________________________________________________________________________________________________________________ 1


18


44440

1111 4324 2432 3243 1342 4133 2221 3414 1423 4212 2144 3331 1234 4441 2313 3122

3232 2443 4311 1124 3421 2214 4142 1333 3344 2131 4223 1412 3113 2322 4434 1241

4343 1132 3224 2411 4114 1321 3433 2242 4231 1444 3312 2123 4422 1213 3141 2334

2424 3211 1143 4332 2233 3442 1314 4121 2112 3323 1431 4244 2341 3134 1222 4413

1223 4412 2344 3131 1434 4241 2113 3322 1311 4124 2232 3443 1142 4333 2421 3214

3144 2331 4423 1212 3313 2122 4234 1441 3432 2243 4111 1324 3221 2414 4342 1133

4431 1244 3112 2323 4222 1413 3341 2134 4143 1332 3424 2211 4314 1121 3233 2442

2312 3123 1231 4444 2141 3334 1422 4213 2224 3411 1343 4132 2433 3242 1114 4321

1334 4141 2213 3422 1123 4312 2444 3231 1242 4433 2321 3114 1411 4224 2132 3343

3413 2222 4134 1341 3244 2431 4323 1112 3121 2314 4442 1233 3332 2143 4211 1424

4122 1313 3441 2234 4331 1144 3212 2423 4414 1221 3133 2342 4243 1432 3324 2111

2241 3434 1322 4113 2412 3223 1131 4344 2333 3142 1214 4421 2124 3311 1443 4232

1442 4233 2121 3314 1211 4424 2332 3143 1134 4341 2413 3222 1323 4112 2244 3431

3321 2114 4242 1433 3132 2343 4411 1224 3213 2422 4334 1141 3444 2231 4123 1312

4214 1421 3333 2142 4443 1232 3124 2311 4322 1113 3241 2434 4131 1344 3412 2223

2133 3342 1414 4221 2324 3111 1243 4432 2441 3234 1122 4313 2212 3423 1331 4144

44440

44440

44440

44440

44440

44440

44440

44440

44440

44440

44440

44440

44440

44440

44440

44440

44440

44440

44440 44440 44440 44440 44440 44440 44440 44440 44440 44440 44440 44440 44440 44440 44440

Its bimagic sum is Sb1616 : 143634120 . Each block of order 4 is a magic square with sum

S44 : 11110 . Result 36. Pan diagonal magic square of order 16 formed by digits 1, 2, 3 and 4 with each block of order 4 also a pan diagonal magic square with sum S44 : 11110 is given by

44440 44440 44440 44440 44440 44440 44440 44440 44440 44440 44440 44440 44440 44440 44440

44440

44440

44440

44440

44440

44440

44440

44440

44440

44440

44440

44440

44440

44440

44440

44440

2333 1222 4342 3213 2432 1123 4443 3112 2134 1421 4141 3414 2231 1324 4244 3311

3242 4313 1233 2322 3143 4412 1132 2423 3441 4114 1434 2121 3344 4211 1331 2224

1213 2342 3222 4333 1112 2443 3123 4432 1414 2141 3421 4134 1311 2244 3324 4231

4322 3233 2313 1242 4423 3132 2412 1143 4121 3434 2114 1441 4224 3331 2211 1344

2234 1321 4241 3314 2131 1424 4144 3411 2433 1122 4442 3113 2332 1223 4343 3212

3341 4214 1334 2221 3444 4111 1431 2124 3142 4413 1133 2422 3243 4312 1232 2323

1314 2241 3321 4234 1411 2144 3424 4131 1113 2442 3122 4433 1212 2343 3223 4332

4221 3334 2214 1341 4124 3431 2111 1444 4422 3133 2413 1142 4323 3232 2312 1243

2431 1124 4444 3111 2334 1221 4341 3214 2232 1323 4243 3312 2133 1422 4142 3413

3144 4411 1131 2424 3241 4314 1234 2321 3343 4212 1332 2223 3442 4113 1433 2122

1111 2444 3124 4431 1214 2341 3221 4334 1312 2243 3323 4232 1413 2142 3422 4133

4424 3131 2411 1144 4321 3234 2314 1241 4223 3332 2212 1343 4122 3433 2113 1442

2132 1423 4143 3412 2233 1322 4242 3313 2331 1224 4344 3211 2434 1121 4441 3114

3443 4112 1432 2123 3342 4213 1333 2222 3244 4311 1231 2324 3141 4414 1134 2421

1412 2143 3423 4132 1313 2242 3322 4233 1211 2344 3224 4331 1114 2441 3121 4434

4123 3432 2112 1443 4222 3333 2213 1342 4324 3231 2311 1244 4421 3134 2414 1141

44440

44440

44440

44440

44440

44440

44440

44440

44440

44440

44440

44440

44440

44440

44440

44440

44440

44440

44440 44440 44440 44440 44440 44440 44440 44440 44440 44440 44440 44440 44440 44440 44440

The magic squares appearing in Results 35 and 36 can be applied to genetic tables [6,8] representing 1, 2, 3 and 4 by letters C, A, T and G respectively.

11.

Magic Squares Order 25

This section deals with magic squares of order 25 in two diffierent forms. One on 7-digits palindormes and second on combinations with repetitions using only the numbers 1, 2, 3 and 4. For 5-digits palindromic magic square of order 25 refer to Taneja [10]. ______________________________________________________________________________________________________________________________________________________________ 1


19


11.1. Palindromic Working with 7-digits palindromes, we have exactly 625 palindromes having the numbers 1, 2, 3, 4 and 5. Based on this, we can write pan diagonal palindromic bimagic square of order 25. Result 37. 7-digits palindromic pan diagonal bimagic square of order 25 having the numbers 1, 2, 3, 4 and 5 is given by Part 1 Part 2 Part 1:

83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325

83333325

83333325

83333325

83333325

83333325

83333325

83333325

83333325

83333325

83333325

83333325

83333325

83333325

83333325

1111111 4422244 2233322 5544455 3355533 5225225 3531353 1342431 4153514 2414142 4334334 2145412 5451545 3212123 1523251 3443443 1254521 4515154 2321232 5132315 2552552 5313135 3124213 1435341 4241424

2244422 5555555 3311133 1122211 4433344 1353531 4114114 2425242 5231325 3542453 5412145 3223223 1534351 4345434 2151512 4521254 2332332 5143415 3454543 1215121 3135313 1441441 4252524 2513152 5324235

3322233 1133311 4444444 2255522 5511155 2431342 5242425 3553553 1314131 4125214 1545451 4351534 2112112 5423245 3234323 5154515 3415143 1221221 4532354 2343432 4213124 2524252 5335335 3141413 1452541

4455544 2211122 5522255 3333333 1144411 3514153 1325231 4131314 2442442 5253525 2123212 5434345 3245423 1551551 4312134 1232321 4543454 2354532 5115115 3421243 5341435 3152513 1413141 4224224 2535352

5533355 3344433 1155511 4411144 2222222 4142414 2453542 5214125 3525253 1331331 3251523 1512151 4323234 2134312 5445445 2315132 5121215 3432343 1243421 4554554 1424241 4235324 2541452 5352535 3113113

4343434 2154512 5415145 3221223 1532351 3452543 1213121 4524254 2335332 5141415 2511152 5322235 3133313 1444441 4255524 1125211 4431344 2242422 5553555 3314133 5234325 3545453 1351531 4112114 2423242

5421245 3232323 1543451 4354534 2115112 4535354 2341432 5152515 3413143 1224221 3144413 1455541 4211124 2522252 5333335 2253522 5514155 3325233 1131311 4442444 1312131 4123214 2434342 5245425 3551553

1554551 4315134 2121212 5432345 3243423 5113115 3424243 1235321 4541454 2352532 4222224 2533352 5344435 3155513 1411141 3331333 1142411 4453544 2214122 5525255 2445442 5251525 3512153 1323231 4134314

2132312 5443445 3254523 1515151 4321234 1241421 4552554 2313132 5124215 3435343 5355535 3111113 1422241 4233324 2544452 4414144 2225222 5531355 3342433 1153511 3523253 1334331 4145414 2451542 5212125

3215123 1521251 4332334 2143412 5454545 2324232 5135315 3441443 1252521 4513154 1433341 4244424 2555552 5311135 3122213 5542455 3353533 1114111 4425244 2231322 4151514 2412142 5223225 3534353 1345431

2525252 5331335 3142413 1453541 4214124 1134311 4445444 2251522 5512155 3323233 5243425 3554553 1315131 4121214 2432342 4352534 2113112 5424245 3235323 1541451 3411143 1222221 4533354 2344432 5155515

3153513 1414141 4225224 2531352 5342435 2212122 5523255 3334333 1145411 4451544 1321231 4132314 2443442 5254525 3515153 5435345 3241423 1552551 4313134 2124212 4544454 2355532 5111115 3422243 1233321

4231324 2542452 5353535 3114113 1425241 3345433 1151511 4412144 2223222 5534355 2454542 5215125 3521253 1332331 4143414 1513151 4324234 2135312 5441445 3252523 5122215 3433343 1244421 4555554 2311132

5314135 3125213 1431341 4242424 2553552 4423244 2234322 5545455 3351533 1112111 3532353 1343431 4154514 2415142 5221225 2141412 5452545 3213123 1524251 4335334 1255521 4511154 2322232 5133315 3444443

1442441 4253524 2514152 5325235 3131313 5551555 3312133 1123211 4434344 2245422 4115114 2421242 5232325 3543453 1354531 3224223 1535351 4341434 2152512 5413145 2333332 5144415 3455543 1211121 4522254

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Part 2: 83333325

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5252525 3513153 1324231 4135314 2441442 4311134 2122212 5433345 3244423 1555551 3425243 1231321 4542454 2353532 5114115 2534352 5345435 3151513 1412141 4223224 1143411 4454544 2215122 5521255 3332333

1335331 4141414 2452542 5213125 3524253 5444445 3255523 1511151 4322234 2133312 4553554 2314132 5125215 3431343 1242421 3112113 1423241 4234324 2545452 5351535 2221222 5532355 3343433 1154511 4415144

2413142 5224225 3535353 1341431 4152514 1522251 4333334 2144412 5455545 3211123 5131315 3442443 1253521 4514154 2325232 4245424 2551552 5312135 3123213 1434341 3354533 1115111 4421244 2232322 5543455

3541453 1352531 4113114 2424242 5235325 2155512 5411145 3222223 1533351 4344434 1214121 4525254 2331332 5142415 3453543 5323235 3134313 1445441 4251524 2512152 4432344 2243422 5554555 3315133 1121211

4124214 2435342 5241425 3552553 1313131 3233323 1544451 4355534 2111112 5422245 2342432 5153515 3414143 1225221 4531354 1451541 4212124 2523252 5334335 3145413 5515155 3321233 1132311 4443444 2254522

3434343 1245421 4551554 2312132 5123215 2543452 5354535 3115113 1421241 4232324 1152511 4413144 2224222 5535355 3341433 5211125 3522253 1333331 4144414 2455542 4325234 2131312 5442445 3253523 1514151

4512154 2323232 5134315 3445443 1251521 3121213 1432341 4243424 2554552 5315135 2235322 5541455 3352533 1113111 4424244 1344431 4155514 2411142 5222225 3533353 5453545 3214123 1525251 4331334 2142412

5145415 3451543 1212121 4523254 2334332 4254524 2515152 5321235 3132313 1443441 3313133 1124211 4435344 2241422 5552555 2422242 5233325 3544453 1355531 4111114 1531351 4342434 2153512 5414145 3225223

1223221 4534354 2345432 5151515 3412143 5332335 3143413 1454541 4215124 2521252 4441444 2252522 5513155 3324233 1135311 3555553 1311131 4122214 2433342 5244425 2114112 5425245 3231323 1542451 4353534

2351532 5112115 3423243 1234321 4545454 1415141 4221224 2532352 5343435 3154513 5524255 3335333 1141411 4452544 2213122 4133314 2444442 5255525 3511153 1322231 3242423 1553551 4314134 2125212 5431345

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83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325 83333325

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Combining Parts 1 and 2 as given above, we get pan diagonal bimagic squares of order 25 with magic sums S2525 : 83333325 and Sb2525 : 328283072727275 with each block of order 5 also a pan diagonal magic squares with sum S55 : 16666665 .

11.2. Combinations with Repetions Allowing repetitions in numbers 1, 2, 3 4 and 5 writing in 4-digits forms, we have exactly 54  625  252 possibilities with repetitions. These values brings a pan diagonal bimagic square of order 25 given in result below. Result 38. Pan diagonal bimagic square of order 25 formed by digits 1, 2, 3, 4 and 5 written 4-digits forms is given by 1111 4422 2233 5544 3355 5225 3531 1342 4153 2414 4334 2145 5451 3212 1523 3443 1254 4515 2321 5132 2552 5313 3124 1435 4241

2244 5555 3311 1122 4433 1353 4114 2425 5231 3542 5412 3223 1534 4345 2151 4521 2332 5143 3454 1215 3135 1441 4252 2513 5324

3322 1133 4444 2255 5511 2431 5242 3553 1314 4125 1545 4351 2112 5423 3234 5154 3415 1221 4532 2343 4213 2524 5335 3141 1452

4455 2211 5522 3333 1144 3514 1325 4131 2442 5253 2123 5434 3245 1551 4312 1232 4543 2354 5115 3421 5341 3152 1413 4224 2535

5533 3344 1155 4411 2222 4142 2453 5214 3525 1331 3251 1512 4323 2134 5445 2315 5121 3432 1243 4554 1424 4235 2541 5352 3113

4343 2154 5415 3221 1532 3452 1213 4524 2335 5141 2511 5322 3133 1444 4255 1125 4431 2242 5553 3314 5234 3545 1351 4112 2423

5421 3232 1543 4354 2115 4535 2341 5152 3413 1224 3144 1455 4211 2522 5333 2253 5514 3325 1131 4442 1312 4123 2434 5245 3551

1554 4315 2121 5432 3243 5113 3424 1235 4541 2352 4222 2533 5344 3155 1411 3331 1142 4453 2214 5525 2445 5251 3512 1323 4134

2132 5443 3254 1515 4321 1241 4552 2313 5124 3435 5355 3111 1422 4233 2544 4414 2225 5531 3342 1153 3523 1334 4145 2451 5212

3215 1521 4332 2143 5454 2324 5135 3441 1252 4513 1433 4244 2555 5311 3122 5542 3353 1114 4425 2231 4151 2412 5223 3534 1345

2525 5331 3142 1453 4214 1134 4445 2251 5512 3323 5243 3554 1315 4121 2432 4352 2113 5424 3235 1541 3411 1222 4533 2344 5155

3153 1414 4225 2531 5342 2212 5523 3334 1145 4451 1321 4132 2443 5254 3515 5435 3241 1552 4313 2124 4544 2355 5111 3422 1233

4231 2542 5353 3114 1425 3345 1151 4412 2223 5534 2454 5215 3521 1332 4143 1513 4324 2135 5441 3252 5122 3433 1244 4555 2311

5314 3125 1431 4242 2553 4423 2234 5545 3351 1112 3532 1343 4154 2415 5221 2141 5452 3213 1524 4335 1255 4511 2322 5133 3444

1442 4253 2514 5325 3131 5551 3312 1123 4434 2245 4115 2421 5232 3543 1354 3224 1535 4341 2152 5413 2333 5144 3455 1211 4522

5252 3513 1324 4135 2441 4311 2122 5433 3244 1555 3425 1231 4542 2353 5114 2534 5345 3151 1412 4223 1143 4454 2215 5521 3332

1335 4141 2452 5213 3524 5444 3255 1511 4322 2133 4553 2314 5125 3431 1242 3112 1423 4234 2545 5351 2221 5532 3343 1154 4415

2413 5224 3535 1341 4152 1522 4333 2144 5455 3211 5131 3442 1253 4514 2325 4245 2551 5312 3123 1434 3354 1115 4421 2232 5543

3541 1352 4113 2424 5235 2155 5411 3222 1533 4344 1214 4525 2331 5142 3453 5323 3134 1445 4251 2512 4432 2243 5554 3315 1121

4124 2435 5241 3552 1313 3233 1544 4355 2111 5422 2342 5153 3414 1225 4531 1451 4212 2523 5334 3145 5515 3321 1132 4443 2254

3434 1245 4551 2312 5123 2543 5354 3115 1421 4232 1152 4413 2224 5535 3341 5211 3522 1333 4144 2455 4325 2131 5442 3253 1514

4512 2323 5134 3445 1251 3121 1432 4243 2554 5315 2235 5541 3352 1113 4424 1344 4155 2411 5222 3533 5453 3214 1525 4331 2142

5145 3451 1212 4523 2334 4254 2515 5321 3132 1443 3313 1124 4435 2241 5552 2422 5233 3544 1355 4111 1531 4342 2153 5414 3225

1223 4534 2345 5151 3412 5332 3143 1454 4215 2521 4441 2252 5513 3324 1135 3555 1311 4122 2433 5244 2114 5425 3231 1542 4353

2351 5112 3423 1234 4545 1415 4221 2532 5343 3154 5524 3335 1141 4452 2213 4133 2444 5255 3511 1322 3242 1553 4314 2125 5431

Magic sums are S2525 : 83325 and Sb2525 : 328227275 with each block of order 5 also a pan diagonal magic squares with sum S55 : 16665 .

Final Comments We have worked with three kinds of multi-digits magic squares. Two of them are connected with combinations of numbers with and without repetitions. The third one is on palindromic numbers. In case of combinations without repetitions, may magic squares are obtained considering order of digits as order of magic squares and in some cases less orders also give interesting magic squares. For order 8, 9, 16 and 25 we have situtions of exact digits in combinations with repetitions as well as in palindromic numbers. See the table below: Orders 8 9 16 25

1,2,3,4 1,2,3 1,2,3,4 1,2,3,4,5

Combinations with repetitions – 3 by 3 => 43=64=82 (Result 17) – 4 by 4 => 34=81=92 (Result 23) – 4 by 4 => 44=256=162 (Results 29 and 30) – 4 by 4 => 54=625=252 (Result 32)

Palindromic numbers 5-digits (1,2,3,4) => 64=82 (Result 16) 7-digits (1,2,3) => 81=92 (Result 22) 7-digits (1,2,3,4) => 256=162 (Results 27 and 28) 7-digits (1,2,3,4,5) => 625=252 (Result 31)

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Still we have exactly 16 palindromes of 7-digits with 1 and 2 forming a pan diagonal magic square of order 4 (Result 5). According Table 2, we have, n4  (n2 )2 . This means, we have combinations with repetitions of n numbers considering 4 by 4 give perfect squares to bring higher order magic squares. In Table 3, we have palindromic possibilities for higher order magic squares. Studies on both these aspects shall be dealt elsewhere. For more studied on magic squares refer [6]-[10].

Acknowledgements Author is thankful to M.A. Amela ([email protected]), T. J. Eckman ([email protected]) and M. Nakamura ([email protected]), for helping in different digits magic squares. Amela helped with magics square of order 4, 6 and 9 (Results 3, 8 and 19). Eckman contributed to magic squares of orders 5 and 7 (Results 6 and 11). Nakamura helped with magic squares of orders 9 and 10 (Results 18 and 24).

References 1. M.A. AMELA, Magic Orthogonal Latin Squares of Order Six, August, 2015, pre-print, pp. 16. 2. M.A. AMELA, Different Digits Magic Square of Order 4, personal communication. 3. T. J. ECKMAN, Different Digits Magic Squares of Orders 5 and 7, personal communication. 4. M. NAKAMURA, Different Digits Magic Squares of Order 10, personal communication. 5.

R.N.Mohan, M.H. Lee and S.S. Pokhrel, On Orthogonality http://arxiv.org/ftp/cs/papers/0604/0604041.pdf

of Latin Squares,

6. I.J. TANEJA, Selfie Palindromic Magic Squares, RGMIA Research Report Collection, 18(2015), Art. 98, pp. 1-15 http://rgmia.org/papers/v18/v18a98.pdf. 7. I.J. TANEJA, Representations of Genetic Tables, Bimagic Squares, Hamming Distances and Shannon Entropy, http://arxiv.org/pdf/1206.2220v1.pdf, June 11, 2012. 8. I.J. TANEJA Intervally Distributed, Palindromic, Selfie Magic Squares, and Double Colored Patterns, RGMIA Research Report Collection, 18(2015), Art. 127, pp. 1-45, http://rgmia.org/papers/v18/v18a127.pdf. 9. I.J. TANEJA, Intervally Distributed, Palindromic and Selfie Magic Squares: Genetic Table and Colored Pattern – Orders 11 to 20, , RGMIA Research Report Collection, 18(2015), Art. 140, pp. 1-43, http://rgmia.org/papers/v18/v18a140.pdf. 10. I.J TANEJA, Intervally Distributed, Palindromic and Selfie Magic Squares of Orders 21 to 25, RGMIA Research Report Collection, 18(2015), Art.151, pp. 133, http://rgmia.org/papers/v18/v18a151.pdf. -----------------------------______________________________________________________________________________________________________________________________________________________________ 1


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