The Games and Puzzles Journal — Issue 32, March-April 2004

This issue is devoted to further new results by Awani Kumar on 12 by 12 magic knight tours. This time he concentrates on reentrant tours that are 4-fold cyclic (i.e. remain magic when renumbered cyclically from the quarter points 37, 73, 109), and also have one diagonal magic.

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Sections on this page: (66) Introduction. (67) Even Diagonal Magic. (68) Even Diagonal Magic with Broken Diagonals. (69) Odd Diagonal Magic with Broken Diagonals. (70) Some Open Tours. End.

## Studies in Magic Tours of Knight on 12*12 Board

### by Awani Kumar

17, Rana Pratap Marg, Lucknow 226001 INDIA
 (66) Introduction é
Tours of Knight have unfathomable mysteries and continue to throw up new results in spite of voluminous literature produced over centuries by untiring works of professional and amateur mathematicians. The author has shown that the very first magic knight's tour on 12*12 board, constructed by Krishnaraj Wadiar, Raja of Mysore, India, before 1868 is a rich mine of re-entrant magic tours. [See The Games and Puzzles Journal # 24]. Further studies have revealed that not only 'simple magic tours' (only rows and columns magic, not diagonals), but an ocean of re-entrant cyclic, 'almost perfect magic tours' (rows, columns and one diagonal adding to magic constant) can be constructed from it.

The author has now enumerated over 100 single diagonal closed magic tours. All these tours are reentrant 4-fold cyclic magic tours, that is, they remain magic when numbered from the quarter points 37, 73 and 109. All the tours 2 to 28 that follow are of this type. The 12*12 board can be divided into 9 quads, each of 4*4 size, and the to and fro path of the knight in all the new tours follows the general plan shown in Fig.1. [Magic diagonals are coloured in the diagrams below. Yellow for even, Orange for odd.]
 Fig.1

 (67) Some Tours with Even Diagonal Magic. é
Fig.2 shows an 'almost perfect magic tour' with its even diagonal adding to magic constant. Fig.3 to Fig.11 can be obtained by modifying quads not lying on the magic diagonal.

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

 (68) Even Magic Diagonal with some Broken Diagonals Magic. é
Fig.12 is also an 'almost perfect magic tour' with its even magic diagonal different from the previous ones. Fig.13 to Fig.15 can be obtained by modifying non magic-diagonal quads. Similarly, Fig.16 has a different arrangement along its magic diagonals. Fig.17 to Fig.20 can be obtained by modifying its non magic-diagonal quads. [Two tours in each set also have an even-numbered broken diagonal which adds to the magic constant.]
 19 54 91 126 21 52 123 94 35 38 107 110 90 127 20 53 124 93 22 51 106 109 36 39 55 18 125 92 49 24 95 122 37 34 111 108 128 89 56 17 96 121 50 23 112 105 40 33 15 58 129 88 25 48 119 100 29 42 113 104 130 87 16 57 120 97 26 45 116 103 32 41 59 14 131 86 47 118 99 28 101 30 43 114 84 133 60 13 98 27 46 117 44 115 102 31 11 62 85 132 7 78 139 68 141 72 1 74 134 83 12 61 138 67 6 77 4 75 142 71 63 10 81 136 65 8 79 140 69 144 73 2 82 135 64 9 80 137 66 5 76 3 70 143
 19 54 91 126 21 52 123 94 35 38 107 110 90 127 20 53 124 93 22 51 106 109 36 39 55 18 125 92 49 24 95 122 37 34 111 108 128 89 56 17 96 121 50 23 112 105 40 33 57 16 87 130 25 48 119 100 29 42 113 104 88 129 58 15 120 97 26 45 116 103 32 41 59 14 131 86 47 118 99 28 101 30 43 114 84 133 60 13 98 27 46 117 44 115 102 31 11 62 85 132 7 78 139 68 141 72 1 74 134 83 12 61 138 67 6 77 4 75 142 71 63 10 81 136 65 8 79 140 69 144 73 2 82 135 64 9 80 137 66 5 76 3 70 143
 19 54 91 126 21 52 123 94 35 38 107 110 90 127 20 53 124 93 22 51 106 109 36 39 55 18 125 92 49 24 95 122 37 34 111 108 128 89 56 17 96 121 50 23 112 105 40 33 15 58 129 88 25 48 119 100 29 42 113 104 130 87 16 57 120 97 26 45 116 103 32 41 59 14 131 86 47 118 99 28 101 30 43 114 84 133 64 9 98 27 46 117 44 115 102 31 13 60 85 132 65 8 79 140 69 144 73 2 134 83 10 63 80 137 66 5 76 3 70 143 61 12 81 136 7 78 139 68 141 72 1 74 82 135 62 11 138 67 6 77 4 75 142 71
 19 54 91 126 21 52 123 94 35 38 107 110 90 127 20 53 124 93 22 51 106 109 36 39 55 18 125 92 49 24 95 122 37 34 111 108 128 89 56 17 96 121 50 23 112 105 40 33 57 16 87 130 25 48 119 100 29 42 113 104 88 129 58 15 120 97 26 45 116 103 32 41 59 14 131 86 47 118 99 28 101 30 43 114 84 133 64 9 98 27 46 117 44 115 102 31 13 60 85 132 65 8 79 140 69 144 73 2 134 83 10 63 80 137 66 5 76 3 70 143 61 12 81 136 7 78 139 68 141 72 1 74 82 135 62 11 138 67 6 77 4 75 142 71
 19 54 125 92 51 22 123 94 35 38 107 110 90 127 20 53 124 93 50 23 106 109 36 39 55 18 91 126 21 52 95 122 37 34 111 108 128 89 16 57 96 121 24 49 112 105 40 33 17 56 129 88 25 48 97 120 29 44 113 104 130 87 58 15 98 119 28 45 102 115 32 41 59 14 131 86 47 26 117 100 43 30 103 114 84 133 60 13 118 99 46 27 116 101 42 31 61 12 85 132 7 78 139 68 141 72 1 74 134 83 10 63 138 67 6 77 4 75 142 71 11 62 81 136 65 8 79 140 69 144 73 2 82 135 64 9 80 137 66 5 76 3 70 143
 19 54 125 92 51 22 123 94 35 38 107 110 126 91 20 53 124 93 50 23 106 109 36 39 55 18 89 128 21 52 95 122 37 34 111 108 90 127 56 17 96 121 24 49 112 105 40 33 15 58 129 88 25 48 97 120 29 44 113 104 130 87 16 57 98 119 28 45 102 115 32 41 59 14 131 86 47 26 117 100 43 30 103 114 84 133 12 61 118 99 46 27 116 101 42 31 13 60 85 132 7 78 139 68 141 72 1 74 134 83 62 11 138 67 6 77 4 75 142 71 63 10 81 136 65 8 79 140 69 144 73 2 82 135 64 9 80 137 66 5 76 3 70 143
 19 54 125 92 51 22 123 94 35 38 107 110 126 91 20 53 124 93 50 23 106 109 36 39 55 18 127 90 21 52 95 122 37 34 111 108 128 89 16 57 96 121 24 49 112 105 40 33 17 56 87 130 25 48 97 120 29 44 113 104 88 129 58 15 98 119 28 45 102 115 32 41 13 60 131 86 47 26 117 100 43 30 103 114 84 133 14 59 118 99 46 27 116 101 42 31 61 12 85 132 7 78 139 68 141 72 1 74 134 83 62 11 138 67 6 77 4 75 142 71 63 10 81 136 65 8 79 140 69 144 73 2 82 135 64 9 80 137 66 5 76 3 70 143
 55 18 125 92 51 22 123 94 35 38 107 110 90 127 54 19 124 93 50 23 106 109 36 39 17 56 91 126 21 52 95 122 37 34 111 108 128 89 20 53 96 121 24 49 112 105 40 33 57 16 87 130 25 48 97 120 29 44 113 104 88 129 58 15 98 119 28 45 102 115 32 41 59 14 131 86 47 26 117 100 43 30 103 114 84 133 12 61 118 99 46 27 116 101 42 31 13 60 85 132 7 78 139 68 141 72 1 74 134 83 62 11 138 67 6 77 4 75 142 71 63 10 81 136 65 8 79 140 69 144 73 2 82 135 64 9 80 137 66 5 76 3 70 143
 19 54 125 92 51 22 123 94 35 38 107 110 126 91 20 53 124 93 50 23 106 109 36 39 55 18 127 90 21 52 95 122 37 34 111 108 128 89 16 57 96 121 24 49 112 105 40 33 17 56 87 130 25 48 97 120 29 44 113 104 88 129 58 15 98 119 28 45 116 101 32 41 13 60 131 86 47 26 117 100 43 30 103 114 84 133 14 59 118 99 46 27 102 115 42 31 61 12 85 132 7 78 139 68 141 72 1 74 134 83 62 11 138 67 6 77 4 75 142 71 63 10 81 136 65 8 79 140 69 144 73 2 82 135 64 9 80 137 66 5 76 3 70 143
Fig.20a

Fig.28 is the nearest to perfection so far achieved. In this the odd diagonal is 44 short of the magic constant. However, its broken odd diagonal, shown in orange, also adds to the magic constant. [Note this tour has been moved out of sequence, but to avoid confusion the editor decided not to renumber all the tours.]
 127 54 19 92 21 94 123 50 35 38 107 110 18 91 126 53 124 51 22 95 106 109 36 39 55 128 89 20 93 24 49 122 37 34 111 108 90 17 56 125 52 121 96 23 112 105 40 33 15 58 129 88 25 48 27 44 117 102 113 104 86 131 16 57 120 97 118 101 28 43 32 41 59 14 87 130 47 26 99 116 45 30 103 114 132 85 60 13 98 119 46 29 100 115 42 31 61 12 133 84 65 8 79 140 69 144 73 2 134 83 64 9 80 137 66 5 76 3 70 143 11 62 81 136 7 78 139 68 141 72 1 74 82 135 10 63 138 67 6 77 4 75 142 71
Figure 28a

 (69) Odd Magic Diagonal with some Broken Diagonals Magic. é
It is true that the reverse of an even diagonal magic tour gives an odd diagonal magic tour but the tour shown in Fig.21 is not related like that to previous tours. It has its odd diagonal magic. Fig.22 to Fig.27 can be derived from it by modifying non magic-diagonal quads.
 19 54 125 92 21 52 123 94 35 38 107 110 126 91 20 53 124 93 22 51 106 109 36 39 55 18 127 90 49 24 95 122 37 34 111 108 88 129 16 57 96 121 50 23 112 105 40 33 17 56 89 128 119 48 25 100 29 42 113 104 130 87 58 15 26 97 120 45 116 103 32 41 13 60 131 86 47 118 99 28 101 30 43 114 132 85 14 59 98 27 46 117 44 115 102 31 61 12 81 8 65 138 77 140 69 144 73 2 84 133 64 137 80 7 68 5 76 3 70 143 11 62 135 82 9 66 139 78 141 72 1 74 134 83 10 63 136 79 6 67 4 75 142 71
 19 54 125 92 21 52 123 94 35 38 107 110 126 91 20 53 124 93 22 51 106 109 36 39 55 18 127 90 49 24 95 122 37 34 111 108 88 129 16 57 96 121 50 23 112 105 40 33 17 56 89 128 119 48 25 100 29 42 113 104 130 87 58 15 26 97 120 45 116 103 32 41 13 60 131 86 47 118 99 28 101 30 43 114 132 85 14 59 98 27 46 117 44 115 102 31 61 12 83 136 65 8 77 140 69 144 73 2 84 133 62 9 80 137 68 5 76 3 70 143 11 82 135 64 7 66 139 78 141 72 1 74 134 63 10 81 138 79 6 67 4 75 142 71
 19 54 125 92 21 52 123 94 35 38 107 110 126 91 20 53 124 93 22 51 106 109 36 39 55 18 127 90 49 24 95 122 37 34 111 108 88 129 16 57 96 121 50 23 112 105 40 33 17 56 89 128 119 48 25 100 29 42 113 104 130 87 58 15 26 97 120 45 116 103 32 41 13 60 131 86 47 118 99 28 101 30 43 114 132 85 14 59 98 27 46 117 44 115 102 31 61 12 83 136 65 6 79 140 69 144 73 2 84 133 62 9 80 139 66 5 76 3 70 143 11 82 135 64 137 78 7 68 141 72 1 74 134 63 10 81 8 67 138 77 4 75 142 71
 19 54 125 92 21 52 123 94 35 38 107 110 126 91 20 53 124 93 22 51 106 109 36 39 55 18 127 90 49 24 95 122 37 34 111 108 88 129 16 57 96 121 50 23 112 105 40 33 17 56 89 128 119 48 25 100 29 42 113 104 130 87 58 15 26 97 120 45 116 103 32 41 13 60 131 86 47 118 99 28 101 30 43 114 132 85 14 59 98 27 46 117 44 115 102 31 61 12 83 136 79 138 67 6 69 144 73 2 84 133 62 9 66 7 78 139 76 3 70 143 11 82 135 64 137 80 5 68 141 72 1 74 134 63 10 81 8 65 140 77 4 75 142 71
 19 54 125 92 21 52 123 94 35 38 107 110 126 91 20 53 124 93 22 51 106 109 36 39 55 18 127 90 49 24 95 122 37 34 111 108 88 129 16 57 96 121 50 23 112 105 40 33 17 56 89 128 119 48 25 100 29 42 113 104 130 87 58 15 26 97 120 45 116 103 32 41 13 60 131 86 47 118 99 28 101 30 43 114 132 85 14 59 98 27 46 117 44 115 102 31 83 12 61 136 65 8 77 140 69 144 73 2 62 133 84 9 80 137 68 5 76 3 70 143 11 82 135 64 7 66 139 78 141 72 1 74 134 63 10 81 138 79 6 67 4 75 142 71
 19 54 125 92 21 52 123 94 35 38 107 110 126 91 20 53 124 93 22 51 106 109 36 39 55 18 127 90 49 24 95 122 37 34 111 108 88 129 16 57 96 121 50 23 112 105 40 33 17 56 89 128 119 48 25 100 29 42 113 104 130 87 58 15 26 97 120 45 116 103 32 41 13 60 131 86 47 118 99 28 101 30 43 114 132 85 14 59 98 27 46 117 44 115 102 31 83 12 61 136 79 138 67 6 69 144 73 2 62 133 84 9 66 7 78 139 76 3 70 143 11 82 135 64 137 80 5 68 141 72 1 74 134 63 10 81 8 65 140 77 4 75 142 71
 19 54 125 92 21 52 123 94 35 38 107 110 126 91 20 53 124 93 22 51 106 109 36 39 55 18 127 90 49 24 95 122 37 34 111 108 88 129 16 57 96 121 50 23 112 105 40 33 17 56 89 128 119 48 25 100 29 42 113 104 130 87 58 15 26 97 120 45 116 103 32 41 13 60 131 86 47 118 99 28 101 30 43 114 132 85 14 59 98 27 46 117 44 115 102 31 83 12 61 136 65 6 79 140 69 144 73 2 62 133 84 9 80 139 66 5 76 3 70 143 11 82 135 64 137 78 7 68 141 72 1 74 134 63 10 81 8 67 138 77 4 75 142 71
Figure 27a
Fig.25 and Fig.27 are unique in the sense that two of their broken diagonals are magic. [The editor notes that in the tour in Fig.27 the magic diagonals occur at regular intervals. Note also that Fig.28 has been moved to the end of section (68).]

Historical note. T. H. Willcocks, G. P. Jelliss and Awani Kumar have previously constructed 'almost perfect magic tours' of open type, but only H. J. R. Murray (1947) had previously constructed, a re-entrant 4-fold cyclic 'almost perfect magic tour' (although he may not have noticed that one diagonal was magic). However, his tour is not 'prolific' in producing other such tours. So, such tours, once rare are now dime a dozen. Human ingenuity can do wonders!

 (70) Some Open Tours. é
T. H. Willcocks has constructed three open, single diagonal magic tours from which the author has derived over 80 such tours along with four diagonally magic tours. [See The Games and Puzzles Journal # 26]. G.P.Jelliss (2003) has also constructed four single diagonal open magic tours from which the author has derived five more such tours as shown in Fig.29 to Fig.33.
 5 64 101 120 3 66 107 114 35 70 111 74 100 121 4 65 106 115 2 67 112 73 34 71 63 6 119 102 61 8 113 108 69 36 75 110 122 99 62 7 116 105 68 1 76 109 72 33 11 58 103 118 9 60 81 140 37 32 77 144 98 123 10 59 104 117 42 27 80 141 38 31 57 12 125 96 43 26 139 82 29 40 143 78 124 97 56 13 138 83 28 41 142 79 30 39 15 54 95 126 25 44 85 136 23 46 87 134 94 127 14 55 84 137 24 45 86 135 22 47 53 16 129 92 51 18 131 90 49 20 133 88 128 93 52 17 130 91 50 19 132 89 48 21
 5 64 105 116 3 66 107 114 35 70 111 74 104 117 4 65 106 115 2 67 112 73 34 71 63 6 119 102 61 8 113 108 69 36 75 110 118 103 62 7 120 101 68 1 76 109 72 33 11 58 99 122 9 60 81 140 37 32 77 144 98 123 10 59 100 121 42 27 80 141 38 31 57 12 129 92 43 26 139 82 29 40 143 78 124 97 44 25 130 91 28 41 142 79 30 39 13 56 93 128 23 46 83 138 21 48 85 136 96 125 24 45 90 131 22 47 84 137 20 49 55 14 127 94 53 16 133 88 51 18 135 86 126 95 54 15 132 89 52 17 134 87 50 19
 5 64 105 116 3 66 107 114 35 70 111 74 104 117 4 65 106 115 2 67 112 73 34 71 63 6 119 102 61 8 113 108 69 36 75 110 118 103 62 7 120 101 68 1 76 109 72 33 11 58 99 122 9 60 81 140 37 32 77 144 124 97 10 59 100 121 42 27 80 141 38 31 57 12 123 98 43 26 139 82 29 40 143 78 96 125 56 13 138 83 28 41 142 79 30 39 55 14 95 126 25 44 85 136 23 46 87 134 94 127 52 17 84 137 24 45 86 135 22 47 15 54 129 92 51 18 131 90 49 20 133 88 128 93 16 53 130 91 50 19 132 89 48 21
 5 64 105 116 3 66 107 114 35 70 111 74 104 117 4 65 106 115 2 67 112 73 34 71 63 6 119 102 61 8 113 108 69 36 75 110 118 103 62 7 120 101 68 1 76 109 72 33 11 58 123 98 9 60 81 140 37 32 77 144 124 97 10 59 100 121 42 27 80 141 38 31 57 12 99 122 43 26 139 82 29 40 143 78 96 125 56 13 138 83 28 41 142 79 30 39 55 14 95 126 25 44 85 136 23 46 87 134 94 127 16 53 84 137 24 45 86 135 22 47 15 54 129 92 51 18 131 90 49 20 133 88 128 93 52 17 130 91 50 19 132 89 48 21
 5 64 105 116 3 66 107 114 35 70 111 74 104 117 4 65 106 115 2 67 112 73 34 71 63 6 119 102 61 8 113 108 69 36 75 110 118 103 62 7 120 101 68 1 76 109 72 33 11 58 123 98 9 60 81 140 37 32 77 144 124 97 10 59 100 121 42 27 80 141 38 31 57 12 99 122 43 26 139 82 29 40 143 78 96 125 56 13 138 83 28 41 142 79 30 39 55 14 95 126 25 44 85 136 23 46 87 134 94 127 52 17 84 137 24 45 86 135 22 47 15 54 129 92 51 18 131 90 49 20 133 88 128 93 16 53 130 91 50 19 132 89 48 21
Conclusion: A re-entrant 'perfect magic tour' (both diagonals magic) has remained elusive. However, broken diagonals adding to the magic constant have been shown in several of the figures. The whole world of pan diagonal magic tours has remained unexplored. Readers are requested to look into it.

Using powerful computers and intelligent programming, the international team of Hugues Mackay, JC Meyrignac, and Guenter Stertenbrink et al has enumerated all the 280 magic tours on 8*8 board. So the work started by Beverley (1848) has been finished after 155 years in 2003. With dedicated and sustained effort, let us enumerate all the 12*12 magic tours within the next 5 years.

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