Block-Wise Magic and Bimagic Squares - RGMIA

Inder J. Taneja

RGMIA Research Report Collection, 20(2017), pp.1-45, http://rgmia.org/v20.php

Received 16/12/17

Block-Wise Magic and Bimagic Squares Inder J. Taneja1 Abstract This paper summarize some of the results done before by author [31, 32, 33, 34, 36] on different ways are writing block-wise magic squares. In this paper, we shall rewrite some these results without details. The details can be seen in above references. This is done for the magic squares of orders 12 to 36, i.e., orders 12, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 33, 35 and 36. In some cases, the magic squares are bimagic or semi-bimagic. In all the cases, at least one of the block-wise representation is pan diagonal except the orders 18 and 30.

Contents 1 Basic Magic Squares 1.1 Magic Square of Order 3 . . . . . . . . . 1.2 Magic Square of Order 4 . . . . . . . . . 1.3 Magic Square of Order 5 . . . . . . . . . 1.4 Magic Square of Order 6 . . . . . . . . . 1.5 Magic Square of Order 7 . . . . . . . . . 1.6 Magic and Bimagic Squares of Order 8 . 1.7 Magic and Bimagic Squares of Order 9 . 1.8 Magic Square of Order 10 . . . . . . . . 1.9 Magic Squares of Order 11 . . . . . . . . 1.10 Magic Square of Order 14 . . . . . . . .

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2 Block-Wise Magic and Bimagic Squares 2.1 Magic Squares of Order 12 . . . . . . . . . . . . . . . . 2.1.1 First Approach: 9 Blocks of Order 4 . . . . . . . 2.1.2 Second Approach: 12 Blocks of Order 3 . . . . . 2.1.3 Third Approach: 4 Blocks of Order 6 . . . . . . . 2.2 Magic Squares of Order 15 . . . . . . . . . . . . . . . . 2.2.1 First Approach: 9 Blocks of Order 5 . . . . . . . 2.2.2 Second Approach: 25 Blocks of Order 3 . . . . . 2.3 Magic and Bimagic Squares of Order 16 . . . . . . . . 2.3.1 First Approach: Pan Magic Square of Order 16 . 2.3.2 Second Approach: Bimagic Square of Order 16 2.4 Magic Square of Order 18 . . . . . . . . . . . . . . . . . 2.4.1 First Approach: 36 Blocks of Order 3 . . . . . . . 2.4.2 Second Approach: 9 Blocks of Order 6 . . . . . .

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Formerly, Professor of Mathematics, Universidade Federal de Santa Catarina, Florianópolis, SC, Brazil (1978-2012). Also worked at Delhi University, India (1976-1978). E-mail: [email protected]; Web-site: http://inderjtaneja.com; Twitter: @IJTANEJA. 1

RGMIA Res.Rep. Coll. 20 (2017), Art. 171, 45 pp

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Inder J. Taneja


2.5 Magic Square of Order 20 . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 First Approach: 25 Blocks of Order 4 . . . . . . . . . . . . . 2.5.2 Second Approach: 16 Blocks of Order 5 . . . . . . . . . . . 2.5.3 Third Approach: 4 Blocks of Order 10 . . . . . . . . . . . . 2.6 Magic Square of Order 21 . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 First Approach: 9 Blocks of Order 7 . . . . . . . . . . . . . 2.6.2 Second Approach: 49 Blocks of Order 3 . . . . . . . . . . . 2.7 Magic and Semi-Bimagic Square of Order 24 . . . . . . . . . . . 2.7.1 First Approach: 36 Blocks of Order 4 . . . . . . . . . . . . . 2.7.2 Second Approach: 64 Blocks of Order 3 . . . . . . . . . . . 2.7.3 Third Approach: 16 Blocks of Order 6 . . . . . . . . . . . . 2.7.4 Forth Approach: Semi-Bimagic Square of Order 24 . . . . 2.8 Bimagic Square of Order 25 . . . . . . . . . . . . . . . . . . . . . . 2.9 Magic Squares of Order 27 . . . . . . . . . . . . . . . . . . . . . . 2.9.1 First Approach: 81 Blocks of Order 3 . . . . . . . . . . . . . 2.9.2 Second Approach: 9 Blocks of Order 9 . . . . . . . . . . . . 2.9.3 Third Approach: 9 Blocks of Bimagic Squares of Order 9 . 2.10 Magic Square of Order 28 . . . . . . . . . . . . . . . . . . . . . . . 2.10.1 First Approach: 49 Blocks of Order 4 . . . . . . . . . . . . . 2.10.2 Second Approach: 16 Blocks of Order 7 . . . . . . . . . . . 2.10.3 Third Approach: 4 Blocks of Order 14 . . . . . . . . . . . . 2.11 Magic Square of Order 30 . . . . . . . . . . . . . . . . . . . . . . . 2.11.1 First Approach: 36 Blocks of Order 5 . . . . . . . . . . . . . 2.11.2 Second Approach: 25 Blocks of Order 6 . . . . . . . . . . . 2.11.3 Third Approach: 9 Blocks of Order 10 . . . . . . . . . . . . 2.11.4 Forth Approach: 100 Blocks of Order 3 . . . . . . . . . . . 2.12 Magic and Bimagic Squares of Order 32 . . . . . . . . . . . . . . 2.12.1 First Approach: 64 Blocks of Order 4 . . . . . . . . . . . . . 2.12.2 Second Approach: 16 Blocks of Order 8 . . . . . . . . . . . 2.13 Magic Squares of Order 33 . . . . . . . . . . . . . . . . . . . . . . 2.13.1 First Approach: 9 Blocks of Order 11 . . . . . . . . . . . . . 2.13.2 Second Approach: 121 Blocks of Order 3 . . . . . . . . . . 2.14 Magic Squares of Order 35 . . . . . . . . . . . . . . . . . . . . . . 2.14.1 First Approach: 49 Blocks of Order 5 . . . . . . . . . . . . . 2.14.2 Second Approach: 25 Blocks of Order 7 . . . . . . . . . . . 2.15 Magic Squares of Order 36 . . . . . . . . . . . . . . . . . . . . . . 2.15.1 First Approach: 81 Blocks of Order 4 . . . . . . . . . . . . . 2.15.2 Second Approach: 9 Blocks of Order 12 . . . . . . . . . . . 2.15.3 Third Approach: 16 Blocks of Order 9 . . . . . . . . . . . . 2.15.4 Forth Approach: 144 Blocks of Order 3 . . . . . . . . . . . 2.15.5 Fifth Approach: 36 Blocks of Order 6 . . . . . . . . . . . . . 2.15.6 Sixth Approach: 16 Blocks of Bimagic Squares of Order 9 .

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3 Final Comments 43 3.1 Author’s Contributions to Magic Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

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1 Basic Magic Squares Below are some magic and bimagic squares well known in the literature. These are of orders, 3, 4, 5, 6,7, 8, 9, 10, 11 and 14. In some cases, these magic squares are pan diagonal, such as of orders 4, 5, 7, 8, 9 and 11. Bimagic squares of orders 8 and 9 are also written. The aim writing these magic squares is that they are used to construct block-wise magic or bimagic squares. 1.1 Magic Square of Order 3 Example 1.1. Let’s consider a magic square of order 3 is given by 15 8 3 4

1 5 9

6 7 2

15 15 15

15 15 15 15 This magic square is used in the block-wise constructions of magic squares of orders 3k, k > 2. 1.2 Magic Square of Order 4 Example 1.2. Let’s consider a pan magic square of order 4. 34 34 34 34 7 12 1 14 34 34 2 13 8 11 34 34 16 3 10 5 34 34 9 6 15 4 34 34 34 34 34 34 This magic square is used in the block-wise constructions of magic squares of orders 4k, k > 1 1.3 Magic Square of Order 5 Example 1.3. Let’s consider a pan magic square of order 5 is given by 65 65 65 65 65 65 65 65 65

1 9 12 20 23 65 17 25 3 6 14 65 8 11 19 22 5 65 24 2 10 13 16 65 15 18 21 4 7 65 65 65 65 65 65 65

This magic square is used in the block-wise constructions of magic squares of orders 5k, k > 2

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1.4 Magic Square of Order 6 Example 1.4. Let’s consider a magic square of order 6. 111 1 30 24 18 7 31

35 8 23 14 26 5

34 28 15 21 10 3

33 9 16 22 27 4

2 11 20 17 29 32

6 25 13 19 12 36

111 111 111 111 111 111

111 111 111 111 111 111 111 This magic square is used in the block-wise constructions of magic squares of orders 6k, k > 1 1.5 Magic Square of Order 7 Example 1.5. Let’s consider a pan magic square of order 7 is given by 175 175 175 175 175 175 175 1 40 23 13 45 35 18

175 175 175 175 175 175

9 48 31 21 4 36 26

17 7 39 22 12 44 34

25 8 47 30 20 3 42

33 16 6 38 28 11 43

41 24 14 46 29 19 2

49 32 15 5 37 27 10

175 175 175 175 175 175 175

175 175 175 175 175 175 175 175 This magic square is used in the block-wise constructions of magic squares of orders 7k, k > 2 1.6 Magic and Bimagic Squares of Order 8 Example 1.6. Let’s consider a pan magic square of order 8 is given by 260 260 260 260 260 260 260 260 260 260 260

25 8 64 33

48 49 9 24

1 32 40 57

56 41 17 16

26 7 63 34

47 50 10 23

2 31 39 58

55 42 18 15

260 260 260 260

260 260 260 260

27 6 62 35

46 51 11 22

3 30 38 59

54 43 19 14

28 5 61 36

45 52 12 21

4 29 37 60

53 44 20 13

260 260 260 260

260 260 260 260 260 260 260 260 260 This magic square is used in the block-wise constructions of magic squares of orders 8k, k > 1 Example 1.7. Let’s consider a pan diagonal bimagic square of order 8 given by 4


Inder J. Taneja

260 260 260 260 260 260 260 260 260

16 26

41 63

36 54

5 19

27 13

62 44

55 33

18 8

260 260

260 260

1 23

40 50

45 59

12 30

22 4

51 37

58 48

31 9

260 260

260 260

38 52

3 21

10 32

47 57

49 39

24 2

29 11

60 46

260 260

260 260

43 61

14 28

7 17

34 56

64 42

25 15

20 6

53 35

260 260

260 260 260 260 260 260 260 260 260 In this case, each 2 × 4 blocks entries sum is same as of magic square, i.e., 260. This magic square is used in the block-wise constructions of bimagic squares of orders 24, 32, etc. In case of order 24 the magic square is semi-bimagic. 1.7 Magic and Bimagic Squares of Order 9 Example 1.8. Let’s consider a pan diagonal magic square of order 9 is given by 369 369 369 369 369 369 369 369 369 369 369

8 37 78

49 63 11

66 23 34

61 12 50

24 35 64

38 76 9

36 65 22

77 7 39

10 51 62

369 369 369

369 369 369

14 52 57

28 69 26

81 2 40

67 27 29

3 41 79

53 55 15

42 80 1

56 13 54

25 30 68

369 369 369

369 369 369

20 31 72

43 75 5

60 17 46

73 6 44

18 47 58

32 70 21

48 59 16

71 19 33

4 45 74

369 369 369

369 369 369 369 369 369 369 369 369 369 Additionally it has property that each 3 × 3 block is of same sum as of magic square, i.e., S 9 = 369. Also each 3 × 3 block is a semi-magic square of order 3 (only in rows and columns). Example 1.9. Let’s consider a bimagic square of order 9 given by 369 1 33 62

18 38 67

23 52 75

35 55 6

40 72 11

48 77 25

60 8 28

65 13 45

79 21 50

369 369 369

27 47 76

5 34 57

10 42 71

49 81 20

30 59 7

44 64 15

74 22 54

61 3 32

69 17 37

369 369 369

14 43 66

19 51 80

9 29 58

39 68 16

53 73 24

31 63 2

70 12 41

78 26 46

56 4 36

369 369 369

369 369 369 369 369 369 369 369 369 369 5


Inder J. Taneja

Each 3 × 3 block is of equal sum entries as of magic square, i.e., 369. The bimagic sum is Sb 9×9 = 20049. The bimagic squares of orders 8 and 9 given in examples 2.20 and 1.9 are the classical onea constructed in 1891 by G. Pfeffermann [13]. 1.8 Magic Square of Order 10 Example 1.10. Let’s consider a magic square of order 10 is given by 505 1 98 47 70 84 13 75 59 26 32

80 12 81 57 99 38 46 24 5 63

65 9 23 88 52 44 40 96 17 71

97 66 79 34 11 10 83 42 58 25

39 90 16 2 45 77 28 61 93 54

22 74 35 91 68 56 19 3 50 87

48 55 94 29 73 82 67 20 31 6

86 33 60 15 7 21 92 78 64 49

53 41 62 76 30 95 4 37 89 18

14 27 8 43 36 69 51 85 72 100

505 505 505 505 505 505 505 505 505 505

505 505 505 505 505 505 505 505 505 505 505 This magic square shall be used to bring 4 blocks of order 10 in magic square of order 20. Also to bring 9 blocks of order 10 in magic square of order 30. 1.9 Magic Squares of Order 11 Example 1.11. Let’s consider a pan diagonal magic square of order 11 is given by 671 671 671 671 671 671 671 671 671 671 671 671 671 671 671 671 671 671 671 671 671

1 13 25 37 49 61 73 85 97 109 121 671 108 120 11 12 24 36 48 60 72 84 96 671 83 95 107 119 10 22 23 35 47 59 71 671 58 70 82 94 106 118 9 21 33 34 46 671 44 45 57 69 81 93 105 117 8 20 32 671 19 31 43 55 56 68 80 92 104 116 7 671 115 6 18 30 42 54 66 67 79 91 103 671 90 102 114 5 17 29 41 53 65 77 78 671 76 88 89 101 113 4 16 28 40 52 64 671 51 63 75 87 99 100 112 3 15 27 39 671 26 38 50 62 74 86 98 110 111 2 14 671 671 671 671 671 671 671 671 671 671 671 671 671

This magic square is used in the block-wise construction of magic square of order 33. 1.10 Magic Square of Order 14 Example 1.12. Let’s consider a magic square of order 14 is given by 6


Inder J. Taneja

1379 1 147 126 190 164 104 19 45 30 68 81 139 177 88

95 16 162 111 84 10 35 65 43 144 187 174 129 124

191 110 31 150 55 89 132 71 60 156 175 8 123 28

112 176 137 46 12 37 57 90 73 27 114 161 192 145

138 188 172 155 121 21 3 36 94 75 69 142 56 109

78 127 63 115 86 166 153 193 182 48 18 108 33 9

58 87 80 5 29 179 196 152 167 105 135 116 20 50

32 117 6 77 101 154 165 181 194 93 52 15 64 128

17 4 47 34 133 195 180 168 151 120 85 79 100 66

159 53 26 170 62 74 92 119 107 136 14 185 141 41

146 70 97 23 173 44 122 130 7 183 106 39 82 157

178 83 99 96 25 134 51 2 118 42 143 61 158 189

125 38 149 140 186 59 72 103 22 11 160 54 91 169

49 163 184 67 148 113 102 24 131 171 40 98 13 76

1379 1379 1379 1379 1379 1379 1379 1379 1379 1379 1379 1379 1379 1379

1379 1379 1379 1379 1379 1379 1379 1379 1379 1379 1379 1379 1379 1379 1379 The inner 4 × 4 block is a pan diagonal magic square of order 4 with magic sum S 4×4 := 694. This magic square shall be used to bring 4 blocks of order 14 in magic square of order 28.

2 Block-Wise Magic and Bimagic Squares In this section, we shall give different ways of writing block-wise magic squares of orders 12 to 36. In some cases, the magic squares are bimagic. Prime orders magic squares, such as, of orders 11, 13, 17, 19, 23, 29 and 31 are excluded as they are not possible to construct in blocks. Also, the magic squares of type 2× primes, such as 14, 22, 26 and 34 are also excluded as in these cases also, it is not possible to construct them as blockwise. Specifically, the magic squares considered are of orders 12, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 33, 35 and 36. These magic squares are already been constructed by author in different works [31, 32, 33, 34]. Here, the aim is to write different possibilities together. The detailed construction can be seen in [31, 32, 33, 34]. 2.1 Magic Squares of Order 12 In this subsection, we shall present three different ways of writing block-wise magic square of order 12. One as 4 × 3, i.e., magic square formed by 9 blocks of pan diagonal magic squares of order 4. Second as 3 × 4, i.e., magic square formed by 16 blocks of magic squares of order 3. Third as 4 blocks of order 6 with equal magic sums. In the first two cases, the magic square of order 12 is pan diagonal, while third case, it is just a magic square. 2.1.1 First Approach: 9 Blocks of Order 4

Example 2.1. 9 blocks of pan diagonal magic squares of order 4 constructed using Example 1.2 lead us to a pan diagonal magic square of order 12 given by

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870 870 870 870 870 870 870 870 870 870 870 870 55 108 1 126 56 107 2 125 57 106 3 124 870 870 18 109 72 91 17 110 71 92 16 111 70 93 870 870 144 19 90 37 143 20 89 38 142 21 88 39 870 870 73 54 127 36 74 53 128 35 75 52 129 34 870 870 58 105 4 123 59 104 5 122 60 103 6 121 870 870 15 112 69 94 14 113 68 95 13 114 67 96 870 870 141 22 87 40 140 23 86 41 139 24 85 42 870 870 76 51 130 33 77 50 131 32 78 49 132 31 870 870 61 102 7 120 62 101 8 119 63 100 9 118 870 870 12 115 66 97 11 116 65 98 10 117 64 99 870 870 138 25 84 43 137 26 83 44 136 27 82 45 870 870 79 48 133 30 80 47 134 29 81 46 135 28 870 870 870 870 870 870 870 870 870 870 870 870 870 870 In this case, the magic sum is S 12×12 = 870. All the 4 × 4 blocks are pan diagonal magic square of order 4 with equal magic sums, S 4×4 = 290. 2.1.2 Second Approach: 12 Blocks of Order 3

Example 2.2. 16 blocks of magic squares of order 3 constructed using Example 1.1 and arranging according to Example 1.2 lead us to a pan diagonal magic square of order 12 given by 870 870 870 870 870 870 870 870 870 870 870 870 870 870

55 69 44

45 56 67

68 43 57

96 106 83 82 95 108 107 84 94

13 3 26

27 14 1

2 25 15

126 112 137 870 136 125 114 870 113 138 124 870

870 870 870

18 4 29

28 17 6

5 30 16

121 111 134 135 122 109 110 133 123

60 70 47

46 59 72

71 48 58

91 105 80 870 81 92 103 870 104 79 93 870

870 132 118 143 870 142 131 120 870 119 144 130

19 9 32

33 20 7

8 31 21

90 100 77 76 89 102 101 78 88

49 63 38

39 50 61

62 37 51

870 870 870

870 870 870

54 64 41

40 53 66

65 42 52

127 117 140 141 128 115 116 139 129

24 10 35

34 23 12

11 36 22

870 870 870

85 75 98

99 86 73

74 97 87

870 870 870 870 870 870 870 870 870 870 870 870 870 We observe that it is not possible to have equal magic sums of order 3 as it is impossible to divide magic sum of order 12 by 4, i.e., 870/4:=217.5. In this case, all the 3 × 3 blocks are magic squares of order 3 with different magic sums giving again a pan diagonal magic square of order 4 given in example below. Example 2.3. The magic sums of 16 blocks of magic squares of order 3 appearing in Example 2.2 given above lead us to a pan diagonal magic square of order 4 given by

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870 870 870 870 168 285 42 375 870 870 51 366 177 276 870 870 393 60 267 150 870 870 258 159 384 69 870 870 870 870 870 870 2.1.3 Third Approach: 4 Blocks of Order 6

Example 2.4. 4 blocks of magic squares of order 6 constructed using Example 1.4 lead us to a magic square of order 12 given by 870 1 137 136 129 8 24 2 138 135 130 7 23 120 32 112 33 41 97 119 31 111 34 42 98 96 89 57 64 80 49 95 90 58 63 79 50 72 56 81 88 65 73 71 55 82 87 66 74 25 104 40 105 113 48 26 103 39 106 114 47 121 17 9 16 128 144 122 18 10 15 127 143

870 870 870 870 870 870

3 139 134 131 6 22 4 140 133 132 5 21 870 118 30 110 35 43 99 117 29 109 36 44 100 870 94 91 59 62 78 51 93 92 60 61 77 52 870 70 54 83 86 67 75 69 53 84 85 68 76 870 27 102 38 107 115 46 28 101 37 108 116 45 870 123 19 11 14 126 142 124 20 12 13 125 141 870 870 870 870 870 870 870 870 870 870 870 870 870 870 In this case, the magic sum is S 12×12 := 870, and all the four blocks of order 6 are magic squares with equal magic sums S 6×6 := 435. 2.2 Magic Squares of Order 15 In this subsection, we shall present two different ways of writing block-wise magic square of order 15. One as 5 × 3, i.e., magic square formed by 9 blocks of pan diagonal magic squares of order 5 with equal magic sums. The second as 3 × 5, i.e., magic square formed by 25 blocks semi-magic squares of order 3 (only in rows and columns) with equal semi-magic sums. 2.2.1 First Approach: 9 Blocks of Order 5

Example 2.5. 9 pan diagonal magic squares of order 5 constructed according to Example 1.3 lead us to a pan diagonal magic square of order 15 given by

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1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695

In this case, the magic sum is S 15×15 = 1695. All the 5 × 5 blocks are pan diagonal magic squares of order 5 with equal magic sums S 5×5 = 565. 2.2.2 Second Approach: 25 Blocks of Order 3

Example 2.6. 25 blocks of order 3 constructed according to Example 1.1 lead us to a pan diagonal magic square of order 15 given by 1695

1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695

1695

1695

1695

1695

1695

1695

1695

1695

1695

1695

1695

1695

1695

186 65 88 187 74 78 188 75 76 189 64 86 190 62 87 80 193 66 89 183 67 90 181 68 79 191 69 77 192 70 73 81 185 63 82 194 61 83 195 71 84 184 72 85 182 36 200 103 37 209 93 38 210 91 39 199 101 40 197 102 95 43 201 104 33 202 105 31 203 94 41 204 92 42 205 208 96 35 198 97 44 196 98 45 206 99 34 207 100 32 6 215 118 7 224 108 8 225 106 9 214 116 10 212 117 110 13 216 119 3 217 120 1 218 109 11 219 107 12 220 223 111 5 213 112 14 211 113 15 221 114 4 222 115 2 156 50 133 157 59 123 158 60 121 159 49 131 160 47 132 125 163 51 134 153 52 135 151 53 124 161 54 122 162 55 58 126 155 48 127 164 46 128 165 56 129 154 57 130 152 171 20 148 172 29 138 173 30 136 174 19 146 175 17 147 140 178 21 149 168 22 150 166 23 139 176 24 137 177 25 28 141 170 18 142 179 16 143 180 26 144 169 27 145 167 1695

1695

1695

1695

1695

1695

1695

1695

1695

1695

1695

1695

1695

1695

1695

1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695

In this case, the magic sum is S 15×15 = 1695. All the 3 × 3 blocks are semi-magic squares of order 3 with equal semi-magic sums, i.e., S 3×3 = 339 (in rows and columns). 10


Inder J. Taneja

2.3 Magic and Bimagic Squares of Order 16 In this subsection, we shall present two different ways of writing magic squares of order 16. One is pan diagonal magic square formed by 16 pan diagonal magic squares of order 4 of equal magic sums. The second is bimagic square of order 16. 2.3.1 First Approach: Pan Magic Square of Order 16

Example 2.7. A pan diagonal magic square of order 16 constructed according to Example 1.2 is given by 2056

2056 2056 2056 2056 2056 2056 2056 2056 2056 2056 2056 2056 2056 2056 2056

2056

2056

2056

2056

2056

2056

2056

2056

2056

2056

2056

2056

2056

2056

97 192 1 224 98 191 2 223 99 190 3 222 100 189 4 221 32 193 128 161 31 194 127 162 30 195 126 163 29 196 125 164 256 33 160 65 255 34 159 66 254 35 158 67 253 36 157 68 129 96 225 64 130 95 226 63 131 94 227 62 132 93 228 61 101 188 5 220 102 187 6 219 103 186 7 218 104 185 8 217 28 197 124 165 27 198 123 166 26 199 122 167 25 200 121 168 252 37 156 69 251 38 155 70 250 39 154 71 249 40 153 72 133 92 229 60 134 91 230 59 135 90 231 58 136 89 232 57 105 184 9 216 106 183 10 215 107 182 11 214 108 181 12 213 24 201 120 169 23 202 119 170 22 203 118 171 21 204 117 172 248 41 152 73 247 42 151 74 246 43 150 75 245 44 149 76 137 88 233 56 138 87 234 55 139 86 235 54 140 85 236 53 109 180 13 212 110 179 14 211 111 178 15 210 112 177 16 209 20 205 116 173 19 206 115 174 18 207 114 175 17 208 113 176 244 45 148 77 243 46 147 78 242 47 146 79 241 48 145 80 141 84 237 52 142 83 238 51 143 82 239 50 144 81 240 49 2056

2056

2056

2056

2056

2056

2056

2056

2056

2056

2056

2056

2056

2056

2056

2056

2056 2056 2056 2056 2056 2056 2056 2056 2056 2056 2056 2056 2056 2056 2056 2056 2056 2056

In this case, the magic sum is S 16×16 = 2056. All the 4 × 4 blocks are pan diagonal magic square of order 4 with equal magic sums, S 4×4 = 514. 2.3.2 Second Approach: Bimagic Square of Order 16

Example 2.8. A bimagic square of order 16 is given by

11


Inder J. Taneja

2056

1 154 239 120 23 144 249 98 44 179 198 93 62 165 212 75 232 127 10 145 242 105 32 135 205 86 35 188 219 68 53 174 122 225 152 15 112 247 130 25 83 204 189 38 69 222 171 52 159 8 113 234 137 18 103 256 182 45 92 195 164 59 78 213 46 181 196 91 60 163 214 77 7 160 233 114 17 138 255 104 203 84 37 190 221 70 51 172 226 121 16 151 248 111 26 129 85 206 187 36 67 220 173 54 128 231 146 9 106 241 136 31 180 43 94 197 166 61 76 211 153 2 119 240 143 24 97 250 55 176 217 66 33 186 207 88 30 133 244 107 12 147 230 125 210 73 64 167 200 95 42 177 251 100 21 142 237 118 3 156 80 215 162 57 90 193 184 47 101 254 139 20 115 236 157 6 169 50 71 224 191 40 81 202 132 27 110 245 150 13 124 227 28 131 246 109 14 149 228 123 49 170 223 72 39 192 201 82 253 102 19 140 235 116 5 158 216 79 58 161 194 89 48 183 99 252 141 22 117 238 155 4 74 209 168 63 96 199 178 41 134 29 108 243 148 11 126 229 175 56 65 218 185 34 87 208 2056

2056

2056

2056

2056

2056

2056

2056

2056

2056

2056

2056

2056

2056

2056

2056

2056 2056 2056 2056 2056 2056 2056 2056 2056 2056 2056 2056 2056 2056 2056 2056 2056

In this case, the magic and bimagic sums are S 4x4 := 2056 and Sb 16x16 := 351576 respectively. Details of this bimagic square can be seen in author’s work [26]. 2.4 Magic Square of Order 18 In this subsection, we shall present two different ways of writing magic square of order 18. One is with 36 blocks of magic squares of order 3 with different magic sums. The second is with 9 blocks of magic squares of order 6 of equal magic sums. 2.4.1 First Approach: 36 Blocks of Order 3

Example 2.9. By the applications of Examples 1.1 and 1.4, we get a magic square of order 18 given by

12


Inder J. Taneja

2925

19 3 38

39 20 1

2 37 21

193 177 212 246 262 227 300 280 317 139 159 122 213 194 175 226 245 264 316 299 282 123 140 157 176 211 195 263 228 244 281 318 298 158 121 141

78 94 59

58 77 96

95 60 76

247 267 230 231 248 265 266 229 249

73 93 56

57 74 91

92 55 75

301 285 320 132 148 113 187 171 206 321 302 283 112 131 150 207 188 169 284 319 303 149 114 130 170 205 189

31 15 50

51 32 13

14 49 33

90 70 107 106 89 72 71 108 88

36 16 53

52 35 18

17 54 34

127 147 110 241 261 224 289 273 308 192 172 209 111 128 145 225 242 259 309 290 271 208 191 174 146 109 129 260 223 243 272 307 291 173 210 190

294 274 311 138 154 119 310 293 276 118 137 156 275 312 292 155 120 136

30 10 47

46 29 12

11 48 28

198 178 215 84 64 101 235 255 218 214 197 180 100 83 66 219 236 253 179 216 196 65 102 82 254 217 237

181 165 200 295 279 314 85 69 104 201 182 163 315 296 277 105 86 67 164 199 183 278 313 297 68 103 87

25 9 44

45 26 7

8 43 27

252 268 233 133 153 116 232 251 270 117 134 151 269 234 250 152 115 135

144 160 125 240 256 221 186 166 203 124 143 162 220 239 258 202 185 168 161 126 142 257 222 238 167 204 184

79 99 62

63 80 97

98 61 81

24 4 41

40 23 6

5 42 22

306 286 323 322 305 288 287 324 304

2925

2925

2925

2925

2925

2925

2925

2925

2925

2925

2925

2925

2925

2925

2925

2925

2925

2925

2925 2925 2925 2925 2925 2925 2925 2925 2925 2925 2925 2925 2925 2925 2925 2925 2925 2925 2925

We observe that it is not possible to have equal magic sums of order 3 as it is impossible to divide magic sum of order 18 by 6, i.e., 2925/6:=487.5. All the 36 blocks are magic squares of order 3 with different magic sums resulting again a magic square of order 6 given in example below. Example 2.10. The magic sums of magic squares of order 3 of Example 2.9 lead us to magic square of order 6 given by 2925 60 744 267 879 546 429

582 222 105 411 888 717

735 906 384 87 258 555

897 393 726 591 78 240

420 564 870 249 753 69

231 96 573 708 402 915

2925 2925 2925 2925 2925 2925

2925 2925 2925 2925 2925 2925 2925 2.4.2 Second Approach: 9 Blocks of Order 6

Example 2.11. 9 blocks of magic squares of order 6 constructed according to Example 1.4 lead us to a magic square of order 18 given by

13


Inder J. Taneja

2925

1 270 216 162 55 271

307 72 199 126 234 37

306 252 127 181 90 19

289 73 144 198 235 36

18 91 180 145 253 288

54 217 109 163 108 324

2 269 215 161 56 272

308 71 200 125 233 38

305 251 128 182 89 20

290 74 143 197 236 35

17 92 179 146 254 287

53 218 110 164 107 323

3 268 214 160 57 273

309 70 201 124 232 39

304 250 129 183 88 21

291 75 142 196 237 34

16 93 178 147 255 286

52 219 111 165 106 322

4 267 213 159 58 274

310 69 202 123 231 40

303 249 130 184 87 22

292 76 141 195 238 33

15 94 177 148 256 285

51 220 112 166 105 321

5 266 212 158 59 275

311 68 203 122 230 41

302 248 131 185 86 23

293 77 140 194 239 32

14 95 176 149 257 284

50 221 113 167 104 320

6 265 211 157 60 276

312 67 204 121 229 42

301 247 132 186 85 24

294 78 139 193 240 31

13 96 175 150 258 283

49 222 114 168 103 319

7 264 210 156 61 277

313 66 205 120 228 43

300 246 133 187 84 25

295 79 138 192 241 30

12 97 174 151 259 282

48 223 115 169 102 318

8 263 209 155 62 278

314 65 206 119 227 44

299 245 134 188 83 26

296 80 137 191 242 29

11 98 173 152 260 281

47 224 116 170 101 317

9 262 208 154 63 279

315 64 207 118 226 45

298 244 135 189 82 27

297 81 136 190 243 28

10 99 172 153 261 280

46 225 117 171 100 316

2925

2925

2925

2925

2925

2925

2925

2925

2925

2925

2925

2925

2925

2925

2925

2925

2925

2925

2925 2925 2925 2925 2925 2925 2925 2925 2925 2925 2925 2925 2925 2925 2925 2925 2925 2925 2925

The above magic square is with magic sum S 18×18 = 2925, and all the 9 blocks of order 6 are magic squares with equal magic sums, S 6×6 := 975. 2.5 Magic Square of Order 20 In this subsection, we shall present three different ways of writing block-wise magic square of order 20. One as 4 × 5, i.e., magic square formed by 25 blocks of equal magic sums of pan diagonal magic squares of order 4. The second as 5 × 4, i.e., magic square formed by 16 blocks of pan diagonal magic squares of order 5 with different magic sums. The third as 4 blocks of equal magic sums of magic squares of order 10. In the first two cases, the magic squares are pan diagonals, while in the third case is just a magic square of order 20. 2.5.1 First Approach: 25 Blocks of Order 4

Example 2.12. 25 blocks of magic squares of order 4 constructed according to Example 1.2 lead us to a pan diagonal magic square of order 20 is given by

14


Inder J. Taneja

4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

151 50 400 201 156 45 395 206 161 40 390 211 166 35 385 216 171 30 380 221

300 301 51 150 295 306 56 145 290 311 61 140 285 316 66 135 280 321 71 130

1 200 250 351 6 195 245 356 11 190 240 361 16 185 235 366 21 180 230 371

350 251 101 100 345 256 106 95 340 261 111 90 335 266 116 85 330 271 121 80

152 49 399 202 157 44 394 207 162 39 389 212 167 34 384 217 172 29 379 222

299 302 52 149 294 307 57 144 289 312 62 139 284 317 67 134 279 322 72 129

2 199 249 352 7 194 244 357 12 189 239 362 17 184 234 367 22 179 229 372

349 252 102 99 344 257 107 94 339 262 112 89 334 267 117 84 329 272 122 79

153 48 398 203 158 43 393 208 163 38 388 213 168 33 383 218 173 28 378 223

298 303 53 148 293 308 58 143 288 313 63 138 283 318 68 133 278 323 73 128

3 198 248 353 8 193 243 358 13 188 238 363 18 183 233 368 23 178 228 373

348 253 103 98 343 258 108 93 338 263 113 88 333 268 118 83 328 273 123 78

154 47 397 204 159 42 392 209 164 37 387 214 169 32 382 219 174 27 377 224

297 304 54 147 292 309 59 142 287 314 64 137 282 319 69 132 277 324 74 127

4 197 247 354 9 192 242 359 14 187 237 364 19 182 232 369 24 177 227 374

347 254 104 97 342 259 109 92 337 264 114 87 332 269 119 82 327 274 124 77

155 46 396 205 160 41 391 210 165 36 386 215 170 31 381 220 175 26 376 225

296 305 55 146 291 310 60 141 286 315 65 136 281 320 70 131 276 325 75 126

5 196 246 355 10 191 241 360 15 186 236 365 20 181 231 370 25 176 226 375

346 255 105 96 341 260 110 91 336 265 115 86 331 270 120 81 326 275 125 76

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010

In this case, the magic sum is S 20×20 = 4010. All the 4 × 4 blocks are pan diagonal magic square of order 4 with the equal magic sums, S 4×4 = 802. 2.5.2 Second Approach: 16 Blocks of Order 5

Example 2.13. 16 blocks of magic squares of order 5 constructed according to Example 1.3 lead us to a pan diagonal magic square of order 20 given by

4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

7 263 119 375 231 2 258 114 370 226 16 272 128 384 240 9 265 121 377 233

135 391 167 23 279 130 386 162 18 274 144 400 176 32 288 137 393 169 25 281

183 39 295 151 327 178 34 290 146 322 192 48 304 160 336 185 41 297 153 329

311 87 343 199 55 306 82 338 194 50 320 96 352 208 64 313 89 345 201 57

359 215 71 247 103 354 210 66 242 98 368 224 80 256 112 361 217 73 249 105

12 268 124 380 236 13 269 125 381 237 3 259 115 371 227 6 262 118 374 230

140 396 172 28 284 141 397 173 29 285 131 387 163 19 275 134 390 166 22 278

188 44 300 156 332 189 45 301 157 333 179 35 291 147 323 182 38 294 150 326

316 92 348 204 60 317 93 349 205 61 307 83 339 195 51 310 86 342 198 54

364 220 76 252 108 365 221 77 253 109 355 211 67 243 99 358 214 70 246 102

1 257 113 369 225 8 264 120 376 232 10 266 122 378 234 15 271 127 383 239

129 385 161 17 273 136 392 168 24 280 138 394 170 26 282 143 399 175 31 287

177 33 289 145 321 184 40 296 152 328 186 42 298 154 330 191 47 303 159 335

305 81 337 193 49 312 88 344 200 56 314 90 346 202 58 319 95 351 207 63

353 209 65 241 97 360 216 72 248 104 362 218 74 250 106 367 223 79 255 111

14 270 126 382 238 11 267 123 379 235 5 261 117 373 229 4 260 116 372 228

142 398 174 30 286 139 395 171 27 283 133 389 165 21 277 132 388 164 20 276

190 46 302 158 334 187 43 299 155 331 181 37 293 149 325 180 36 292 148 324

318 94 350 206 62 315 91 347 203 59 309 85 341 197 53 308 84 340 196 52

366 222 78 254 110 363 219 75 251 107 357 213 69 245 101 356 212 68 244 100

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010

We observe that it is not possible to have equal magic sums of order 5 as it is impossible to divide magic sum 4010 by 4, i.e., 4010/4:=1002.5. In this case, all the 5×5 blocks are magic squares of order 5 with different magic sums giving a pan diagonal magic square of order 4 given in example below. Example 2.14. A pan diagonal magic square of order 4, formed by magic sums of 16 magic squares of order 5 is given by

15


Inder J. Taneja

4010 4010 4010 4010 995 1020 965 1030 4010 4010 970 1025 1000 1015 4010 4010 1040 975 1010 985 4010 4010 1005 990 1035 980 4010 4010 4010 4010 4010 4010 2.5.3 Third Approach: 4 Blocks of Order 10

Example 2.15. 4 blocks of equal sums magic squares of order 10 constructed according to Example 1.10 lead us to a magic square of order 20 given by 4010

1 152 244 80 388 329 176 93 305 237 21 132 264 60 368 349 196 113 285 217

177 64 236 333 81 248 5 389 160 312 197 44 216 353 101 268 25 369 140 292

72 256 85 229 17 141 320 168 393 324 52 276 105 209 37 121 300 188 373 344

384 325 180 148 233 317 92 61 249 16 364 345 200 128 213 297 112 41 269 36

336 100 392 301 169 73 224 257 8 145 356 120 372 281 189 53 204 277 28 125

88 397 13 165 304 232 149 340 76 241 108 377 33 185 284 212 129 360 56 261

309 161 157 396 332 20 253 225 84 68 289 181 137 376 352 40 273 205 104 48

153 228 321 4 260 385 77 316 172 89 133 208 341 24 280 365 57 296 192 109

240 9 308 252 65 96 381 144 337 173 220 29 288 272 45 116 361 124 357 193

245 313 69 97 156 164 328 12 221 400 265 293 49 117 136 184 348 32 201 380

2 151 243 79 387 330 175 94 306 238 22 131 263 59 367 350 195 114 286 218

178 63 235 334 82 247 6 390 159 311 198 43 215 354 102 267 26 370 139 291

71 255 86 230 18 142 319 167 394 323 51 275 106 210 38 122 299 187 374 343

383 326 179 147 234 318 91 62 250 15 363 346 199 127 214 298 111 42 270 35

335 99 391 302 170 74 223 258 7 146 355 119 371 282 190 54 203 278 27 126

87 398 14 166 303 231 150 339 75 242 107 378 34 186 283 211 130 359 55 262

310 162 158 395 331 19 254 226 83 67 290 182 138 375 351 39 274 206 103 47

154 227 322 3 259 386 78 315 171 90 134 207 342 23 279 366 58 295 191 110

239 10 307 251 66 95 382 143 338 174 219 30 287 271 46 115 362 123 358 194

246 314 70 98 155 163 327 11 222 399 266 294 50 118 135 183 347 31 202 379

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010

4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010 4010

The above magic square is with magic sum S 20×20 = 4010, and all the four blocks of order 10 are magic squares with equal magic sums S 10×10 := 2005. 2.6 Magic Square of Order 21 In this subsection, we shall present two different ways of writing block-wise pan diagonal magic square of order 21. One as 7×3, i.e., magic square formed by 9 blocks of pan diagonal magic squares of order 7. Second as 3 × 7, i.e., magic square formed by 49 blocks semi-magic squares of order 3 with equal semi-magic sums (only in rows and columns). In both the cases the magic squares are pan diagonal 2.6.1 First Approach: 9 Blocks of Order 7

Example 2.16. A pan diagonal magic square of order 21 formed by 9 blocks of equal sums magic squares of order 7 constructed according to Example 1.5 is given by

16


Inder J. Taneja

4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

1 370 237 123 386 271 159 43 349 216 102 407 292 138 22 328 195 81 428 313 180

111 396 260 166 12 358 244 90 417 281 145 54 337 223 69 438 302 187 33 316 202

155 19 369 232 118 384 270 134 61 348 211 97 405 291 176 40 327 190 76 426 312

243 106 391 258 165 8 376 222 85 412 279 144 50 355 201 64 433 300 186 29 334

265 153 18 365 250 117 379 286 132 60 344 229 96 400 307 174 39 323 208 75 421

375 239 124 390 253 160 6 354 218 103 411 274 139 48 333 197 82 432 295 181 27

397 264 148 13 363 249 113 418 285 127 55 342 228 92 439 306 169 34 321 207 71

3 371 236 122 385 272 158 45 350 215 101 406 293 137 24 329 194 80 427 314 179

110 395 259 167 11 360 245 89 416 280 146 53 339 224 68 437 301 188 32 318 203

154 20 368 234 119 383 269 133 62 347 213 98 404 290 175 41 326 192 77 425 311

242 108 392 257 164 7 377 221 87 413 278 143 49 356 200 66 434 299 185 28 335

266 152 17 364 251 116 381 287 131 59 343 230 95 402 308 173 38 322 209 74 423

374 238 125 389 255 161 5 353 217 104 410 276 140 47 332 196 83 431 297 182 26

398 263 150 14 362 248 112 419 284 129 56 341 227 91 440 305 171 35 320 206 70

2 372 235 121 387 273 157 44 351 214 100 408 294 136 23 330 193 79 429 315 178

109 394 261 168 10 359 246 88 415 282 147 52 338 225 67 436 303 189 31 317 204

156 21 367 233 120 382 268 135 63 346 212 99 403 289 177 42 325 191 78 424 310

241 107 393 256 163 9 378 220 86 414 277 142 51 357 199 65 435 298 184 30 336

267 151 16 366 252 115 380 288 130 58 345 231 94 401 309 172 37 324 210 73 422

373 240 126 388 254 162 4 352 219 105 409 275 141 46 331 198 84 430 296 183 25

399 262 149 15 361 247 114 420 283 128 57 340 226 93 441 304 170 36 319 205 72

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641

The above magic square is with magic sum S 21×21 = 4641, and all the nine blocks of order 7 are pan diagonal magic squares with equal magic sums S 7×7 := 1547. 2.6.2 Second Approach: 49 Blocks of Order 3

Example 2.17. A pan diagonal magic square of order 21 formed by 49 blocks of semi-magic squares of order 3 constructed according to Examples 1.1 and 1.5 is given by

4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

246 300 117 288 6 369 183 384 96 225 27 411 162 426 75 267 48 348 204 321 138

111 243 309 363 285 15 90 180 393 405 222 36 69 159 435 342 264 57 132 201 330

306 120 237 12 372 279 390 99 174 33 414 216 432 78 153 54 351 258 327 141 195

232 312 119 274 18 371 169 396 98 211 39 413 148 438 77 253 60 350 190 333 140

123 245 295 375 287 1 102 182 379 417 224 22 81 161 421 354 266 43 144 203 316

308 106 249 14 358 291 392 85 186 35 400 228 434 64 165 56 337 270 329 127 207

250 299 114 292 5 366 187 383 93 229 26 408 166 425 72 271 47 345 208 320 135

110 240 313 362 282 19 89 177 397 404 219 40 68 156 439 341 261 61 131 198 334

303 124 236 9 376 278 387 103 173 30 418 215 429 82 152 51 355 257 324 145 194

233 314 116 275 20 368 170 398 95 212 41 410 149 440 74 254 62 347 191 335 137

125 242 296 377 284 2 104 179 380 419 221 23 83 158 422 356 263 44 146 200 317

305 107 251 11 359 293 389 86 188 32 401 230 431 65 167 53 338 272 326 128 209

252 298 113 294 4 365 189 382 92 231 25 407 168 424 71 273 46 344 210 319 134

109 239 315 361 281 21 88 176 399 403 218 42 67 155 441 340 260 63 130 197 336

302 126 235 8 378 277 386 105 172 29 420 214 428 84 151 50 357 256 323 147 193

234 311 118 276 17 370 171 395 97 213 38 412 150 437 76 255 59 349 192 332 139

122 244 297 374 286 3 101 181 381 416 223 24 80 160 423 353 265 45 143 202 318

307 108 248 13 360 290 391 87 185 34 402 227 433 66 164 55 339 269 328 129 206

247 301 115 289 7 367 184 385 94 226 28 409 163 427 73 268 49 346 205 322 136

112 241 310 364 283 16 91 178 394 406 220 37 70 157 436 343 262 58 133 199 331

304 121 238 10 373 280 388 100 175 31 415 217 430 79 154 52 352 259 325 142 196

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641

4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641 4641

In this case, the magic sum is S 21×21 = 4641. All the 3 × 3 blocks are semi-magic squares with equal semimagic sums, S 3×3 = 663 (in rows and columns).

17


Inder J. Taneja

2.7 Magic and Semi-Bimagic Square of Order 24 In this subsection, we shall present three different ways of writing block-wise magic square of order 24. One as 4 × 6, i.e., magic square formed by 36 blocks of pan diagonal magic squares of order 4 with equal magic sums. Second as 3×8, i.e., magic square formed by 64 blocks of magic squares of order 3 with different magic sums. The third as 6 × 4, i.e., 16 blocks of magic squares of order 6 with equal magic sums. In the first two cases, the magic squares of order 24 are pan diagonal magic squares, while in the third case it is just a magic square. 2.7.1 First Approach: 36 Blocks of Order 4

Example 2.18. A pan diagonal magic square of order 24 formed by 36 blocks of pan diagonal magic squares of order 4 constructed according to Example 1.2 is given by

6924 6924 6924 6924 6924 6924 6924 6924 6924 6924 6924 6924 6924 6924 6924 6924 6924 6924 6924 6924 6924 6924 6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

217 72 576 289

432 433 73 216

1 288 360 505

504 361 145 144

218 71 575 290

431 434 74 215

2 287 359 506

503 362 146 143

219 70 574 291

430 435 75 214

3 286 358 507

502 363 147 142

220 69 573 292

429 436 76 213

4 285 357 508

501 364 148 141

221 68 572 293

428 437 77 212

5 284 356 509

500 365 149 140

222 67 571 294

427 438 78 211

6 283 355 510

499 366 150 139

6924

223 66 570 295

426 439 79 210

7 282 354 511

498 367 151 138

224 65 569 296

425 440 80 209

8 281 353 512

497 368 152 137

225 64 568 297

424 441 81 208

9 280 352 513

496 369 153 136

226 63 567 298

423 442 82 207

10 279 351 514

495 370 154 135

227 62 566 299

422 443 83 206

11 278 350 515

494 371 155 134

228 61 565 300

421 444 84 205

12 277 349 516

493 372 156 133

6924

229 60 564 301

420 445 85 204

13 276 348 517

492 373 157 132

230 59 563 302

419 446 86 203

14 275 347 518

491 374 158 131

231 58 562 303

418 447 87 202

15 274 346 519

490 375 159 130

232 57 561 304

417 448 88 201

16 273 345 520

489 376 160 129

233 56 560 305

416 449 89 200

17 272 344 521

488 377 161 128

234 55 559 306

415 450 90 199

18 271 343 522

487 378 162 127

6924

235 54 558 307

414 451 91 198

19 270 342 523

486 379 163 126

236 53 557 308

413 452 92 197

20 269 341 524

485 380 164 125

237 52 556 309

412 453 93 196

21 268 340 525

484 381 165 124

238 51 555 310

411 454 94 195

22 267 339 526

483 382 166 123

239 50 554 311

410 455 95 194

23 266 338 527

482 383 167 122

240 49 553 312

409 456 96 193

24 265 337 528

481 384 168 121

6924

241 48 552 313

408 457 97 192

25 264 336 529

480 385 169 120

242 47 551 314

407 458 98 191

26 263 335 530

479 386 170 119

243 46 550 315

406 459 99 190

27 262 334 531

478 387 171 118

244 45 549 316

405 460 100 189

28 261 333 532

477 388 172 117

245 44 548 317

404 461 101 188

29 260 332 533

476 389 173 116

246 43 547 318

403 462 102 187

30 259 331 534

475 390 174 115

6924

247 42 546 319

402 463 103 186

31 258 330 535

474 391 175 114

248 41 545 320

401 464 104 185

32 257 329 536

473 392 176 113

249 40 544 321

400 465 105 184

33 256 328 537

472 393 177 112

250 39 543 322

399 466 106 183

34 255 327 538

471 394 178 111

251 38 542 323

398 467 107 182

35 254 326 539

470 395 179 110

252 37 541 324

397 468 108 181

36 253 325 540

469 396 180 109

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924 6924 6924

6924 6924 6924

6924 6924 6924

6924 6924 6924

6924 6924 6924

6924 6924 6924

In this case, the magic sum is S 24×24 := 6924. All the 36 blocks of order 4 are pan diagonal magic squares with equal magic sums, S 4×4 := 1154. 2.7.2 Second Approach: 64 Blocks of Order 3

Example 2.19. A pan diagonal magic square of order 24 formed by 64 blocks of pan diagonal magic squares of order 3 constructed according to Examples 1.1 and 1.6 is given by

18


Inder J. Taneja

6924 6924 6924 6924 6924 6924 6924 6924 6924 6924 6924 6924 6924 6924 6924 6924 6924 6924 6924 6924 6924 6924 6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

253 279 230

231 254 277

278 229 255

336 310 359

358 335 312

311 360 334

25 3 50

51 26 1

2 49 27

540 562 515

514 539 564

563 516 538

258 280 233

232 257 282

281 234 256

331 309 356

357 332 307

308 355 333

30 4 53

52 29 6

5 54 28

535 561 512

513 536 559

560 511 537

6924

36 10 59

58 35 12

11 60 34

529 555 506

507 530 553

554 505 531

264 286 239

238 263 288

287 240 262

325 303 350

351 326 301

302 349 327

31 9 56

57 32 7

8 55 33

534 556 509

508 533 558

557 510 532

259 285 236

237 260 283

284 235 261

330 304 353

352 329 306

305 354 328

6924

552 574 527

526 551 576

575 528 550

37 15 62

63 38 13

14 61 39

324 298 347

346 323 300

299 348 322

241 267 218

219 242 265

266 217 243

547 573 524

525 548 571

572 523 549

42 16 65

64 41 18

17 66 40

319 297 344

345 320 295

296 343 321

246 268 221

220 245 270

269 222 244

6924

313 291 338

339 314 289

290 337 315

252 274 227

226 251 276

275 228 250

541 567 518

519 542 565

566 517 543

48 22 71

70 47 24

23 72 46

318 292 341

340 317 294

293 342 316

247 273 224

225 248 271

272 223 249

546 568 521

520 545 570

569 522 544

43 21 68

69 44 19

20 67 45

6924

181 159 206

207 182 157

158 205 183

408 430 383

382 407 432

431 384 406

97 123 74

75 98 121

122 73 99

468 442 491

490 467 444

443 492 466

186 160 209

208 185 162

161 210 184

403 429 380

381 404 427

428 379 405

102 124 77

76 101 126

125 78 100

463 441 488

489 464 439

440 487 465

6924

108 130 83

82 107 132

131 84 106

457 435 482

483 458 433

434 481 459

192 166 215

214 191 168

167 216 190

397 423 374

375 398 421

422 373 399

103 129 80

81 104 127

128 79 105

462 436 485

484 461 438

437 486 460

187 165 212

213 188 163

164 211 189

402 424 377

376 401 426

425 378 400

6924

480 454 503

502 479 456

455 504 478

109 135 86

87 110 133

134 85 111

396 418 371

370 395 420

419 372 394

169 147 194

195 170 145

146 193 171

475 453 500

501 476 451

452 499 477

114 136 89

88 113 138

137 90 112

391 417 368

369 392 415

416 367 393

174 148 197

196 173 150

149 198 172

6924

385 411 362

363 386 409

410 361 387

180 154 203

202 179 156

155 204 178

469 447 494

495 470 445

446 493 471

120 142 95

94 119 144

143 96 118

390 412 365

364 389 414

413 366 388

175 153 200

201 176 151

152 199 177

474 448 497

496 473 450

449 498 472

115 141 92

93 116 139

140 91 117

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924 6924

6924 6924

6924 6924

6924 6924

6924 6924

6924 6924

6924 6924

6924 6924

We observe that it is not possible to have equal magic sums of order 3 as it is impossible to divide magic sum of order 24 by 8, i.e., 6924/8:=865.5. In this case, each 3 × 3 block is a magic square of order 3 with different magic sums resulting in a pan diagonal magic square of order 8 given in example below. Example 2.20. A pan diagonal magic square of order 8 arising due to magic sums of order 3 of Exemple 2.19 is given by 6924 6924 6924 6924 6924 6924 6924 6924 762 105

1005 1590

78 789

1617 978

771 96

996 1599

87 780

1608 6924 987 6924

6924 1653 6924 942

114 753

969 1626

726 141

1644 951

123 744

960 1635

735 132

6924 6924

546 321

1221 1374

294 573

1401 1194

555 312

1212 1383

303 564

1392 6924 1203 6924

6924 1437 6924 1158

330 537

1185 1410

510 357

1428 1167

339 528

1176 1419

519 348

6924

6924 6924

6924 6924

6924 6924 6924 6924 6924 6924 6924 6924 6924 2.7.3 Third Approach: 16 Blocks of Order 6

Example 2.21. A magic square of order 24 formed by 16 blocks of equal sums magic squares of order 6 is given by

19


Inder J. Taneja

6924

1 480 384 288 97 481

545 128 353 224 416 65

544 448 225 321 160 33

513 129 256 352 417 64

32 161 320 257 449 512

96 385 193 289 192 576

2 479 383 287 98 482

546 127 354 223 415 66

543 447 226 322 159 34

514 130 255 351 418 63

31 162 319 258 450 511

95 386 194 290 191 575

3 478 382 286 99 483

547 126 355 222 414 67

542 446 227 323 158 35

515 131 254 350 419 62

30 163 318 259 451 510

94 387 195 291 190 574

4 477 381 285 100 484

548 125 356 221 413 68

541 445 228 324 157 36

516 132 253 349 420 61

29 164 317 260 452 509

93 388 196 292 189 573

6924

5 476 380 284 101 485

549 124 357 220 412 69

540 444 229 325 156 37

517 133 252 348 421 60

28 165 316 261 453 508

92 389 197 293 188 572

6 475 379 283 102 486

550 123 358 219 411 70

539 443 230 326 155 38

518 134 251 347 422 59

27 166 315 262 454 507

91 390 198 294 187 571

7 474 378 282 103 487

551 122 359 218 410 71

538 442 231 327 154 39

519 135 250 346 423 58

26 167 314 263 455 506

90 391 199 295 186 570

8 473 377 281 104 488

552 121 360 217 409 72

537 441 232 328 153 40

520 136 249 345 424 57

25 168 313 264 456 505

89 392 200 296 185 569

6924

9 472 376 280 105 489

553 120 361 216 408 73

536 440 233 329 152 41

521 137 248 344 425 56

24 169 312 265 457 504

88 393 201 297 184 568

10 471 375 279 106 490

554 119 362 215 407 74

535 439 234 330 151 42

522 138 247 343 426 55

23 170 311 266 458 503

87 394 202 298 183 567

11 470 374 278 107 491

555 118 363 214 406 75

534 438 235 331 150 43

523 139 246 342 427 54

22 171 310 267 459 502

86 395 203 299 182 566

12 469 373 277 108 492

556 117 364 213 405 76

533 437 236 332 149 44

524 140 245 341 428 53

21 172 309 268 460 501

85 396 204 300 181 565

6924

13 468 372 276 109 493

557 116 365 212 404 77

532 436 237 333 148 45

525 141 244 340 429 52

20 173 308 269 461 500

84 397 205 301 180 564

14 467 371 275 110 494

558 115 366 211 403 78

531 435 238 334 147 46

526 142 243 339 430 51

19 174 307 270 462 499

83 398 206 302 179 563

15 466 370 274 111 495

559 114 367 210 402 79

530 434 239 335 146 47

527 143 242 338 431 50

18 175 306 271 463 498

82 399 207 303 178 562

16 465 369 273 112 496

560 113 368 209 401 80

529 433 240 336 145 48

528 144 241 337 432 49

17 176 305 272 464 497

81 400 208 304 177 561

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924 6924 6924 6924 6924

6924 6924 6924 6924 6924

6924 6924 6924 6924 6924

6924 6924 6924 6924 6924

The above magic square is with magic sum S 24×24 = 6924, and all the 16 blocks of order 6 are magic squares with equal magic sums S 6×6 := 1731. 2.7.4 Forth Approach: Semi-Bimagic Square of Order 24

Below is a magic square of order 24 with 9 blocks of equal sums magic squares of order 8. The magic square is semi-bimagic square of order 24. Example 2.22. A semi-bimagic square of order 24 formed by 9 blocks of equal sums magic squares of order 8 constructed according to Examples 2.20 is given by 6924

142 268 3 165 304 442 417 567

411 573 310 436 9 159 136 274

298 448 423 561 124 286 21 147

15 153 130 280 429 555 292 454

273 135 160 10 435 309 574 412

568 418 441 303 166 4 267 141

453 291 556 430 279 129 154 16

148 22 285 123 562 424 447 297

119 245 26 188 329 467 392 542

386 548 335 461 32 182 113 251

323 473 398 536 101 263 44 170

38 176 107 257 404 530 317 479

248 110 185 35 458 332 551 389

545 395 464 326 191 29 242 116

476 314 533 407 254 104 179 41

173 47 260 98 539 401 470 320

96 222 49 211 354 492 367 517

361 523 360 486 55 205 90 228

348 498 373 511 78 240 67 193

61 199 84 234 379 505 342 504

223 85 210 60 481 355 528 366

522 372 487 349 216 54 217 91

499 337 510 384 229 79 204 66

198 72 235 73 516 378 493 343

6924

95 221 50 212 353 491 368 518

362 524 359 485 56 206 89 227

347 497 374 512 77 239 68 194

62 200 83 233 380 506 341 503

224 86 209 59 482 356 527 365

521 371 488 350 215 53 218 92

500 338 509 383 230 80 203 65

197 71 236 74 515 377 494 344

144 270 1 163 306 444 415 565

409 571 312 438 7 157 138 276

300 450 421 559 126 288 19 145

13 151 132 282 427 553 294 456

271 133 162 12 433 307 576 414

570 420 439 301 168 6 265 139

451 289 558 432 277 127 156 18

150 24 283 121 564 426 445 295

118 244 27 189 328 466 393 543

387 549 334 460 33 183 112 250

322 472 399 537 100 262 45 171

39 177 106 256 405 531 316 478

249 111 184 34 459 333 550 388

544 394 465 327 190 28 243 117

477 315 532 406 255 105 178 40

172 46 261 99 538 400 471 321

6924

120 246 25 187 330 468 391 541

385 547 336 462 31 181 114 252

324 474 397 535 102 264 43 169

37 175 108 258 403 529 318 480

247 109 186 36 457 331 552 390

546 396 463 325 192 30 241 115

475 313 534 408 253 103 180 42

174 48 259 97 540 402 469 319

94 220 51 213 352 490 369 519

363 525 358 484 57 207 88 226

346 496 375 513 76 238 69 195

63 201 82 232 381 507 340 502

225 87 208 58 483 357 526 364

520 370 489 351 214 52 219 93

501 339 508 382 231 81 202 64

196 70 237 75 514 376 495 345

143 269 2 164 305 443 416 566

410 572 311 437 8 158 137 275

299 449 422 560 125 287 20 146

14 152 131 281 428 554 293 455

272 134 161 11 434 308 575 413

569 419 440 302 167 5 266 140

452 290 557 431 278 128 155 17

149 23 284 122 563 425 446 296

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924

6924 6924 6924 6924 6924 6924 6924

6924 6924 6924 6924 6924 6924 6924

6924 6924 6924 6924 6924 6924 6924

In this case, the magic square is formed by 9 blocks of magic squares of order 8 with equal magic sums S 8×8 := 2308. Out of 9 blocks of order 8, three of them are bimagic squares with different bimagic sums, and the other 6 are semi-bimagic (bimagic only in rows and columns) resulting in a semi-bimagic square of order 24 (bimagic only in rows and columns), with semi-bimagic sums are Sb124×24 := 2661124 (rows and columns), Sb224×24 := 2654292 (upper diagonal) and Sb324×24 := 2714116 (lower diagonal). 20

Inder J. Taneja


2.8 Bimagic Square of Order 25 In this subsection, we shall present a block-wise pan diagonal magic square of order 25 with 25 blocks of pan diagonal magic square of order 5. The magic square is also bimagic. Example 2.23. A pan diagonal magic square of order 25 formed by 25 blocks of pan diagonal magic squares of order 5 constructed according to Example 1.3 is given by

In this case the pan diagonal magic square is bimagic with magic and bimagic sums, S 25×25 := 7825 and Sb 25×25 := 3263025 respectively. Each block of order 5 is a pan diagonal magic square with equal magic sums, S 5×5 := 1565. Details of this bimagic square can be seen in author’s work [27]. 2.9 Magic Squares of Order 27 In this subsection, we shall present three different ways of writing block-wise pan diagonal magic square of order 27. One as 3 × 9, i.e., magic square formed by 81 blocks of equal sums semi-magic squares of order 3. Second as 9 × 3, i.e., magic square formed by 9 blocks of pan diagonal magic squares of order 9. Third as, 9 blocks of bimagic squares of order 9 resulting a magic square of order 27. 2.9.1 First Approach: 81 Blocks of Order 3

Example 2.24. A pan diagonal magic square of order 27 formed by 81 blocks of semi-magic squares of order 3 with equal semi-magic sums is given by

21

Inder J. Taneja


In this case, the magic sum is S 27×27 := 9855. All the 81 blocks of order 3 are semi-magic with equal semimagic sums, S 3×3 := 1095 (semi-magic in rows and columns). 2.9.2 Second Approach: 9 Blocks of Order 9

Example 2.25. A pan diagonal magic square of order 27 formed by 9 blocks of pan diagonal magic squares of order 9 with equal magic sums is given by

22

Inder J. Taneja


In this case, the magic sums is S 27×27 := 9855. All the 9 blocks of order 9 are pan diagonal magic squares of equal magic sums, S 9×9 := 3285. The construction of this magic square is due to Dwane [9]. 2.9.3 Third Approach: 9 Blocks of Bimagic Squares of Order 9

Example 2.26. A magic square of order 27 formed by 9 blocks of bimagic squares of order 9 is given by

23

Inder J. Taneja


In this case, the magic square of order 27 is not a bimagic, but it is formed by 9 blocks of different magic and bimagic sums of order 9 resulting in magic square of order 3 of magic square sums of order 9. Example 2.27. The magic square sums of order 9 give us a magic square of order 3 given by 9855 3276 3537 3042 9855 3051 3285 3519 9855 3528 3033 3294 9855 9855 9855 9855 9855 Unfortunately, still we don’t have a proper block-wise bimagic or semi-magic square of order 27. 2.10 Magic Square of Order 28 In this subsection, we shall present three different ways of writing pan diagonal magic square of order 28. One as 4 × 7, i.e., magic square formed by 49 blocks of pan diagonal magic squares of order 4 with equal magic sums. Second as 7 × 4, i.e., magic square formed by 16 blocks of pan diagonal magic squares of order 7. Third as, 14 × 2, i.e., magic square formed by 4 blocks of equal magic sums of order 14. In the first two cases the magic squares of order 28 are pan diagonal, while in the third case, it is just a magic square. 2.10.1 First Approach: 49 Blocks of Order 4


24

Inder J. Taneja


In this case, all the 49 blocks of order 4 are pan diagonal magic squares with equal magic sums, S 4×4 := 1570 2.10.2 Second Approach: 16 Blocks of Order 7


25


Inder J. Taneja

We observe that it is not possible to have equal magic sums of order 7 as it is impossible to divide magic square sum of order 28 by 4, i.e., 10990/4 := 2747.5. In this case, all the 16 blocks of order 7 are pan diagonal magic squares with different magic sums resulting in a pan diagonal magic square of order 4 given in example below. Example 2.30. A pan diagonal magic square of order 4, formed by 16 magic square sums of each 7 × 7 block is given by. 10990 10990 10990 10990 10990

2737 2702

2772 2779

2695 2744

2786 2765

10990 10990

10990 10990

2800 2751

2709 2730

2758 2793

2723 2716

10990 10990

10990 10990 10990 10990 10990 2.10.3 Third Approach: 4 Blocks of Order 14

Example 2.31. A pan diagonal magic square of order 28 formed by 4 blocks of magic squares of order 14 constructed according to Example 1.12 is given by

26

Inder J. Taneja


In this case, all the 4 blocks of order 14 are magic squares with equal magic sums S 14×14 := 5495. The middle blocks in all the magic squares of order 14 are pan diagonal magic squares of order 4 with equal magic sums S 4×4 := 2770. 2.11 Magic Square of Order 30 In this subsection, we shall present four different ways of writing block-wise magic square of order 30. One as 5×6, i.e., magic square formed by 36 blocks of pan diagonal magic squares of order 5. Second as 6×5, i.e., magic square formed by 25 blocks of magic squares of order 6. The third as 10 × 3, i.e., magic square formed by 9 blocks of magic squares of order 10. The forth way is 3 × 10, i.e., magic square formed by 100 blocks of magic squares of order 3 with different magic sums. 2.11.1 First Approach: 36 Blocks of Order 5

Example 2.32. A magic square of order 30 formed by 36 blocks of pan diagonal magic squares of order 5 constructed according to Example 1.3 is given by

27


Inder J. Taneja

We observe that it is not possible to have equal magic sums of order 5 as it is impossible to divide magic square sum of order 30 by 6, i.e., 13515/6 := 2252.5. In this case, all the 36 blocks are pan diagonal magic squares of order 5 with different magic sums, resulting in a magic square of order 6 given in example below. Example 2.33. Magic sums of 36 blocks of pan diagonal magic squares of order 5 lead us to a magic square of order 6 given by 13515 2165 2305 2220 2320 2255 2250

2275 2195 2190 2240 2325 2290

2300 2335 2225 2180 2215 2260

2330 2230 2295 2280 2175 2205

2245 2265 2315 2210 2310 2170

2200 2185 2270 2285 2235 2340

13515 13515 13515 13515 13515 13515

13515 13515 13515 13515 13515 13515 13515 2.11.2 Second Approach: 25 Blocks of Order 6

Example 2.34. A magic square of order 30 formed by 25 blocks of magic squares of order 6 constructed according to Example 1.4 is given by

28

Inder J. Taneja


In this case, all the 25 blocks of order 6 are magic squares with equal magic sums, S 6×6 := 2703. 2.11.3 Third Approach: 9 Blocks of Order 10


29

Inder J. Taneja


In this case, all the 9 blocks of order 10 are magic squares with equal magic sums, S 10×10 := 4505. 2.11.4 Forth Approach: 100 Blocks of Order 3


30


Inder J. Taneja

We observe that it is not possible to have equal magic sums of order 3 as it is impossible to divide magic square sum of order 30 by 10, i.e., 13515/10 := 1351.5. In this case, all the 100 blocks are magic squares of order 3 with different magic sums resulting in a magic square of order 10: Example 2.37. Magic sums of 100 blocks of magic squares of order 3 lead us to a magic square of order 10 is given by 13515

96 2031 1185 1797 1473 2292 699 960 2544 438

2598 375 1779 150 906 1203 1464 672 2067 2301

1221 2589 654 2022 1788 366 987 2283 1500 105

1725 1194 2337 933 2040 708 141 2526 402 1509

429 1527 951 636 1212 2580 2265 168 1743 2004

2013 978 420 2274 645 1491 2562 1257 159 1716

1482 2256 2058 1239 2571 177 1770 384 915 663

2310 123 1446 2535 447 1734 1248 2049 681 942

717 1752 2553 411 114 969 1986 1455 2328 1230

924 690 132 1518 2319 1995 393 1761 1176 2607

13515

13515

13515

13515

13515

13515

13515

13515

13515

13515

13515 13515 13515 13515 13515 13515 13515 13515 13515 13515 13515

2.12 Magic and Bimagic Squares of Order 32 In this subsection, we shall present two ways of writing block-wise magic square of order 32. One as 4×8, i.e., , magic square formed by 64 blocks of pan diagonal magic squares of order 4 with equal magic sums. Second as 8 × 4, i.e., 16 blocks of bimagic squares of order 8 with equal magic sums resulting in bimagic square of order 32. 31

Inder J. Taneja


2.12.1 First Approach: 64 Blocks of Order 4


In this case, all the magic squares of order 4 are pan diagonal and with equal magic sums, S 4×4 := 2050 2.12.2 Second Approach: 16 Blocks of Order 8

Example 2.39. A bimagic square of order 32 formed by 16 blocks of magic squares of order 8 constructed according to Example 2.20 is given by

32


Inder J. Taneja

In this case, the magic square is pan diagonal bimagic with magic and bimagic sums, S 32×32 := 16400 and Sb 32×32 := 11201200 respectively. All the block of order 8 are magic squares with equal magic sums, S 8×8 := 4100. Moreover each block of order 8 is semi-bimagic resulting in a pan diagonal magic square of order 4: Example 2.40. A pan diagonal magic square of order 4 formed by 16 semi-bimagic sums of order 8 is given by 11201200 11201200 11201200 11201200 11201200 11201200 11201200

2808468 2857652 2759300 2775780

2775684 2759396 2857620 2808500

2857700 2808452 2775732 2759316

2759348 2775700 2808548 2857604

11201200 11201200 11201200 11201200

11201200 11201200 11201200 11201200 11201200 Sum of semi-diagonals also forms a pan diagonal magic square of order 4. 2.13 Magic Squares of Order 33 In this subsection, we shall present two different ways of writing block-wise magic square of order 33. One as 11x3, i.e., magic square formed by 9 blocks of pan diagonal magic squares of order 11 with equal magic sums. Second as 3 × 11, i.e., 121 blocks of magic squares of order 3 with equal magic sums. 2.13.1 First Approach: 9 Blocks of Order 11

Example 2.41. A pan diagonalmagic square of order 33 formed by 9 blocks of pan diagonal magic squares of order 11 constructed according to Example 1.11 is given by

33

Inder J. Taneja


In this case all the 9 blocks are of pan diagonal magic squares of order 11 with equal magic sums, S 11×11 := 5995 2.13.2 Second Approach: 121 Blocks of Order 3

Example 2.42. A pan diagonalmagic square of order 33 formed by 121 blocks of semi-magic squares of order 3 constructed according to Example 1.1 is given by

34

Inder J. Taneja


In this case all the 121 blocks are of semi-magic squares of order 3 with equal semi-magic sums,S 3×3 := 1635 (in rows and columns). 2.14 Magic Squares of Order 35 In this subsection, we shall present two different ways of writing block-wise magic square of order 35. One as 5 × 7, i.e., magic square formed by 49 blocks of pan diagonal magic squares of order 7 with equal magic sums. Second as 7 × 5, i.e., magic square formed by 25 blocks of pan diagonal magic squares of order 5 with equal magic sums. In both the cases, the magic square of order 35 is pan diagonal. 2.14.1 First Approach: 49 Blocks of Order 5


In this case, all the 49 blocks are pan diagonal magic squares of order 5 with equal magic sums, S 5×5 := 3065 2.14.2 Second Approach: 25 Blocks of Order 7


35

Inder J. Taneja


In this case, all the 25 blocks are of pan diagonal magic squares of order 7 with equal magic sums, S 7×7 := 4291 2.15 Magic Squares of Order 36 In this subsection, we shall present six different ways of writing block-wise magic square of order 36. One as 4 × 9, i.e., magic square formed by 81 blocks of pan diagonal magic squares of order 4. Second as 12 × 3, i.e., magic square formed by 9 blocks of pan diagonal magic squares of order 12. Third, 9 × 4, i.e., magic square formed by 16 blocks of magic squares of order 3. Forth as, 3x12, i.e., magic square formed by 144 blocks of magic squares of order 3. Fifth as, 6 × 6, i.e., magic square formed by 36 blocks of magic squares of order 6 with equal magic sums. Sixed as 4 × 9, i.e., 16 blocks of bimagic squares of order 9. In the first four cases, the magic square of order 36 is pan diagonal, while in the fifht and sixth it is just a magic square of order 36. 2.15.1 First Approach: 81 Blocks of Order 4


36

Inder J. Taneja


In this case, all the blocks of order 4 are pan diagonal magic square with equal magic sums, S 4×4 := 2594. 2.15.2 Second Approach: 9 Blocks of Order 12


37

Inder J. Taneja


In this case, all the blocks of order 12 are pan diagonal magic squares with equal magic sums, S 12×12 := 7782. The 3 × 3 blocks are semi-magic squares with different semi-magic sums of order 3 (in rows and columns). 2.15.3 Third Approach: 16 Blocks of Order 9

Example 2.47. A pan diagonal magic square of order 36 formed by 16 blocks of magic squares of order 9 constructed according to Example 1.8 is given by

38


Inder J. Taneja

In this case, all the 9 × 9 blocks are pan diagonal magic squares of order 9 with different magic square sums. In each case, the 3 × 3 blocks are semi-magic squares of order 3 with equal semi-magic sums (in rows and columns). The magic sums of order 9 pan diagonal magic squares lead us to a pan diagonal magic square of order 4 given in example below: Example 2.48. Magic sums of pan diagonal magic squares of order 9 lead us to a pan diagonal magic of order 4 given by 23346 23346 23346 23346 23346 23346 23346

5823 5778 5904 5841

5868 5877 5787 5814

5769 5832 5850 5895

5886 5859 5805 5796

23346 23346 23346 23346

23346 23346 23346 23346 23346 2.15.4 Forth Approach: 144 Blocks of Order 3

We observe that in the Examples 2.46 and 2.47 that the 3 × 3 blocks are not magic squares of order 3. In Example 2.47, the 3 × 3 blocks are semi-magic squares in each block of order 9 × 9. On the other side, it is impossible to have all the 3 × 3 blocks of equal sums as 23346 is not divisible by 12, i.e., 23346/12 := 1945.5. Below is a example of a pan diagonal magic square of order 36 where each block of order 3 is a magic square giving total 144 blocks of magic squares of order 3: Example 2.49. A pan diagonal magic square of order 36 formed by 144 blocks of magic squares of order 3 constructed according to Example 1.1 is given by 39


Inder J. Taneja

In this case, each block of order 3 is a magic square with different magic sums, resulting in a pan diagonal magic of order 12 given in example below. Example 2.50. 16 blocks of magic squares of order 3 constructed using Example 1.1, we get a pan diagonal magic square of order 12 given by 23346

23346 23346 23346 23346 23346 23346 23346 23346 23346 23346 23346

23346

23346

23346

23346

23346

23346

23346

1464 1158 1797 2481 2787 2148 1806 1473 1140 2139 2472 2805 1149 1788 1482 2796 2157 2463

438 132 771

780 447 114

123 762 456

3399 3057 3714 3705 3390 3075 3066 3723 3381

23346

3354 3048 3687 1509 1167 1824 2436 2778 2121 3696 3363 3030 1815 1500 1185 2130 2445 2760 3039 3678 3372 1176 1833 1491 2769 2112 2454

23346

483 141 798

789 474 159

23346

150 807 465

3453 3111 3768 3759 3444 3129 3120 3777 3435

23346

492 186 825

23346

834 501 168

177 816 510

23346

2427 2733 2094 1410 1104 1743 2085 2418 2751 1752 1419 1086 2742 2103 2409 1095 1734 1428

23346 23346

23346 23346 23346 23346 23346

2382 2724 2067 1455 1113 1770 3408 3102 3741 2076 2391 2706 1761 1446 1131 3750 3417 3084 2715 2058 2400 1122 1779 1437 3093 3732 3426

537 195 852

843 528 213

204 861 519

23346

23346

23346

23346

23346

23346

23346

23346

23346

23346

23346

23346

23346

23346

23346 23346

The above magic square of order 36 is also formed by 16 blocks of magic square of order 9 resulting in a pan magic square of order 4: 40


Inder J. Taneja

Example 2.51. Magic sums of pan diagonal magic squares of order 9 of Exemple 2.49 lead us to a pan diagonal magic of order 4 given by

23346 23346 23346

23346

23346

23346

23346

4419 1422

7416 10089

1341 4500

10170 7335

23346

10332 7173

1503 4338

7254 10251

4257 1584

23346

23346

23346

23346

23346

23346

23346

23346

2.15.5 Fifth Approach: 36 Blocks of Order 6


The 36 blocks of order 6 are magic squares of equal magic sums, S 6×6 := 3891. 2.15.6 Sixth Approach: 16 Blocks of Bimagic Squares of Order 9

Example 2.53. A magic square of order 36 formed by 16 blocks of bimagic squares of order 9 with different magic sums constructed according to Example 1.9 is given by

41


Inder J. Taneja

The 16 blocks of bimagic squares of order 9 brings a pan diagonal magic square of order 4 given in example below. Example 2.54. Magic sums of pan diagonal magic squares of order 9 of Exemple 2.49 lead us to a pan diagonal magic of order 4 given by 23346

23346 23346 23346

23346

23346

23346

5679 6012 5337 6318 5346 6309 5688 6003

23346

6336 5355 5994 5661 5985 5670 6327 5364

23346

23346

23346

23346

23346

23346

23346

23346

Unfortunately, still we are unable to bring bimagic square of order 36, only the order 9 magic squares are bimagic.

3 Final Comments This paper summarize some of the results done before by author on different ways are writing block-wise magic squares. Here, we have rewritten some of them. The construction details can be seen at [31, 32, 33, 34, 36] . This is done for the magic squares of orders 12 to 36, i.e. of orders 12, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 33, 35 and 36. In some cases, the magic squares are pan diagonal or bimagic or semi-bimagic. We observed that in case of magic squares of type 4k, it is always possible to write a pan diagonal magic square with equal magic sums of pan diagonal magic squares of order 4. This we have done the cases, of orders 8, 12, 16, 20, 24, 28, 32 and 36. Up to order 16 this type of construction is also developed by Candy [5]. 42

Inder J. Taneja


In case of orders 6k, it is also possible to write block-wise magic squares with sum magic squares of order 6. This we have done for the magic squares of orders 12, 18, 24, 30 and 36. In case of orders 3k, we found two types of situations. One as blocks of equal sums semi-magic squares of order 3 (in rows and columns). This we have done for the magic squares of orders 12, 18, 24, 30 and 36. The second as blocks of magic squares of order 3 with different magic sums. This we have done for the magic squares of orders 15, 21, 27 and 33. Other block-wise situations for the orders 5, 7, 8, 9, 10, 11, 12 and 14 are also considered. The bimagic squares of orders 8 and 9 given in examples 2.20 and 1.9 are the classical onea constructed in 1891 by G. Pfeffermann [13]. The block-wise bimagic squares of orders 16 and 25 is given by author in [22, 26]. The block-wise bimagic square of order 32 and semi-bimagic square of order 24 can be seen in Taneja [22]. A good collection of bimagic squares are given in a site by C. Boyer [8]. A mathematical study of bimagic squares of orders 16 and 25 is given in [6]. Some ideas of block-wise magic squares can be seen in [5, 15]. Programming calculations of block-wise magic squares up to order 20 are well explained in a site by Danielsson [11]. During past years the author worked with magic squares in different situations. These are given in details below: 3.1 Author’s Contributions to Magic Squares The item-wise author’s work on magic squares is as follows: (i) Digital numbers magic squares - [16, 17, 18, 19, 20, 21]; (ii) Block-wise construction of bimagic squares - [22]; (iii) Connections with genetic tables and Shannon’s entropy - [23]; (iv) Selfie and palindromic-type magic squares - [24]; (v) Intervally distributed and block-wise magic squares - [25, 26, 27]; (vi) Multi-digits magic squares - [28]; (vii) Perfect square sum magic squares with uniformity, minimum sum and Pythagorean triples- [29, 30]; (viii) Block-wise equal and unequal sums magic squares of orders 3k, 4k and 6k - [31, 32, 33, 36]; (ix) Magic rectangles in construction of block-wise pan magic squares - [34]; (x) Magic crosses: repeated and non repeated entries - [35]; (xi) Representations of letters and numbers With equal sums magic squares of orders 4 and 6 - [37, 38].

References [1] Aale de Winkel, Online discussion, The Magic Encyclopedia, http://magichypercubes.com/Encyclopedia/. [2] Alan W. Grogono, Grogono Magic Squares, http://www.grogono.com/magic/ [3] W.H. Benson and O. Jacoby, New Recreations with Magic Squares, New York, Dover Publications, 1976. [4] C. Boyer - Multimagic Squares, http://www.multimagie.com. [5] A. L. Candy, Pandiagonal Magic Squares of Composite Order, Lincoln, Neb., 1941. [6] H. Derksen, C. Eggrmont and A. Essen, Multimagic Squares, The American Mathematical Monthly, Vol. 114, N 8, October 2007, pages 703-713. Also in http://arxiv.org/abs/math.CO/0504083. 43

Inder J. Taneja


[7] H. Heinz - Magic Squares, Magic Stars and Other Patterns, http://www.magicsquares.net. [8] Bogdan Golunski, NUMBER GALAXY, http://www.number-galaxy.eu/. [9] Dwane H. Campbell and Keith A. Campbell, WELCOME TO MAGIC CUBE GENERATOR, http://magictesseract.com. [10] M. Nakamura, Magic Cubes and Tesseracts, http://magcube.la.coocan.jp/magcube/en/rectangles.htm [11] H. Danielsson, Onlin discussion, Construction methos for magic squares, http://www.magicsquares.info/construction.html [12] Francis Gaspalou, Magic Squares, http://www.gaspalou.fr/magic-squares/ ˘ ZEsprit ´ [13] G. Pfeffermann, Les Tablettes du Chercheur, Journal des Jeux dâA et de Combinaisons, (fortnightly magazine) issues of 1891 Paris. [14] Walter Trump, Notes on Magic Squares, http://www.trump.de/magic-squares/index.html. [15] Willem Barink, The construction of perfect panmagic squares of order 4k, (k ≥ 2), www.allergrootste.com/friends/wba/magic-squares.html [16] I.J. Taneja, Digital Era: Magic Squares and 8th May 2010 (08.05.2010), May, 2010, pp. 1-4, https://arxiv.org/abs/1005.1384 - https://goo.gl/XpyvWu. [17] I.J. Taneja, Universal Bimagic Squares and the day 10th October 2010 (10.10.10), Oct, 2010, pp. 1-5, https://arxiv.org/abs/1010.2083 - https://goo.gl/TtrP9B. [18] I.J. Taneja, DIGITAL ERA: Universal Bimagic Squares, Oct, 2010, pp. 1-8, https://arxiv.org/abs/1010.2541; https://goo.gl/MQWgiw. [19] I.J. Taneja, Upside Down Numerical Equation, Bimagic Squares, and the day September 11, Oct. 2010, pp. 1-7, https://arxiv.org/abs/1010.4186; https://goo.gl/kdBEbk. [20] I.J. Taneja, Equivalent Versions of "Khajuraho" and "Lo-Shu" Magic Squares and the day 1st October 2010 (01.10.2010), Nov. 2010, pp. 1-7, https://arxiv.org/abs/1011.0451; https://goo.gl/vnJxoX. [21] I.J. Taneja, Upside Down Magic, Bimagic, Palindromic Squares and Pythagoras Theorem on a Palindromic Day - 11.02.2011, Feb. 2011, pp.1-9, https://arxiv.org/abs/1102.2394; https://goo.gl/dPLezL. [22] I.J. Taneja, Bimagic Squares of Bimagic Squares and an Open Problem, Feb. 2011, pp. 1-14, https://arxiv.org/abs/1102.3052; https://goo.gl/4fuvqs. [23] I.J. Taneja, Representations of Genetic Tables, Bimagic Squares, Hamming Distances and Shannon Entropy, Jun. 2012, pp. 1-19, https://arxiv.org/abs/1206.2220; https://goo.gl/Jd4JXc. [24] 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 - https://goo.gl/n3mhe5.

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Inder J. Taneja


[25] 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 - https://goo.gl/yzcRWa. [26] 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 - https://goo.gl/DE1iyK. [27] I.J. Taneja, Intervally Distributed, Palindromic and Selfie Magic Squares – Orders 21 to 25 , 18(2015), Art. 151, pp. 1-33. http://rgmia.org/papers/v18/v18a151.pdf - https://goo.gl/rzJYuG. [28] I.J. Taneja, Multi-Digits Magic Squares, RGMIA Research Report Collection, 18(2015), Art. 159, pp. 1-22. http://rgmia.org/papers/v18/v18a159.pdf - https://goo.gl/rw13Dw. [29] I.J. Taneja, Magic Squares with Perfect Square Number Sums, Research Report Collection, 20(2017), Article 11, pp. 1-24, http://rgmia.org/papers/v20/v20a11.pdf - https://goo.gl/JFLEZJ. [30] I.J. Taneja, Pythagorean Triples and Perfect Square Sum Magic Squares, RGMIA Research Report Collection, 20(2017), Art. 128, pp. 1-22, http://rgmia.org/papers/v20/v20a128.pdf - https://goo.gl/qUPV66. [31] I.J. Taneja, Block-Wise Equal Sums Pan Magic Squares of Order 4k, RGMIA Research Report Collection, 20(2017), Art. 150, pp. 1-18, http://rgmia.org/papers/v20/v20a150.pdf ; https://goo.gl/DjfTQd. [32] I.J. TANEJA, Block-Wise Equal Sums Magic Squares of Order 3k, RGMIA Research Report Collection, 20(2017), Art. 154, pp. 1-53, http://rgmia.org/papers/v20/v20a154.pdf ; https://goo.gl/UdZPP9. [33] I.J. TANEJA, Block-Wise Unequal Sums Magic Squares, RGMIA Research Report Collection, 20(2017), Art. 155, pp. 1-44, http://rgmia.org/papers/v20/v20a155.pdf ; https://goo.gl/vWUnSB. [34] I.J. TANEJA, Magic Rectangles in construction of Block-Wise Pan Magic Squares, RGMIA Research Report Collection, 20(2017), Art. 159, pp. 1-47, http://rgmia.org/papers/v20/v20a159.pdf ; https://goo.gl/WSC6gr. [35] I.J. TANEJA, Magic Crosses: Repeated and Non Repeated Entries, RGMIA Research Report Collection, 20(2017), Art. 162, pp. 1-36, http://rgmia.org/papers/v20/v20a162.pdf ; https://goo.gl/eNgkoT. [36] I.J. TANEJA, Block-Wise Equal Sums Magic Squares of Orders 6k, RGMIA Research Report Collection, 20(2017), Art. 163, pp. 1-14, http://rgmia.org/papers/v20/v20a163.pdf ; https://goo.gl/ZxUVmF. [37] I.J. TANEJA, Representations of Letters and Numbers With Equal Sums Magic Squares of Order 4, RGMIA Research Report Collection, 20(2017), Art. 166, pp. 1-38, http://rgmia.org/papers/v20/v20a166.pdf ; https://goo.gl/dQe52o. [38] I.J. TANEJA, Representations of Letters and Numbers With Equal Sums Magic Squares of Order 6, RGMIA Research Report Collection, 20(2017), Art. 167, pp. 1-41, http://rgmia.org/papers/v20/v20a167.pdf ; https://goo.gl/utSiYY. ————————————————-

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