The Games and Puzzles Journal — Issue 26, March-April 2003

Awani Kumar has constructed the first 12×12 fully magic knight's tours (both diagonals magic). The details of his work leading to this important result are given in full. Some work by the editor on magic tours on 4m × 4n+2 rectangles follows, first using knight and rook moves, second relaxing the magic condition to ‘quasi-magic’.

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Sections on this page: (36) In Search of 12×12 Magic Square Tours. (37) Four Perfect Magic Tours 12×12. (38) A New Type of Magic Tour. (39) Emperor Magic Tours. (40) Quasi-Magic Tours. End
 In Search of Perfect Magic Tours of Knight on 12×12 Board. by Awani Kumar 17 Rana Pratap Marg, Lucknow 226001. INDIA. (This article was sent by e-mail on 25 March 2003, tours 9 and 27 were added 3 April 2003.) é

Magic squares and knight's tours have been attracting attention of mathematicians for centuries, however, the twain rarely meet. In spite of the very large number of magic squares and knight's tours possible on doubly even boards, until now 16×16 was the smallest board size on which a ‘perfect magic tour’ (with both diagonals adding to the magic constant) were found. The first, by Helga Em. de Vasa, were published in 1962 [See Knight's Tour Notes: 16×16 Magic Tours].

All the 266 magic knight tours found on the 8×8 board are ‘weak’ magic squares, since their diagonals do not add to the magic constant. On the 12×12 board, Murray, Willcocks, Jelliss and Awani Kumar have constructed scores of ‘simple magic tours’. These composers have also constructed ‘almost perfect magic tours’ (with one diagonal adding to the magic constant) but until now a ‘perfect magic tour’ has remained elusive.

Figure 1 by T. H. Willcocks (published in Recreational Mathematics Magazine 1962) is the nearest to perfection previously achieved. A close look at it reveals that the board can be divided into nine ‘quads’, each of size 4×4. The sum of rows and columns of each quad is 290, that is, one-third of the magic constant. The four 2×2 ‘blocks’ within each quad also sum up to 290. The numbers can be divided into four groups: 1 to 36, 37 to 72, 73 to 108 and 109 to 144. Each of the thirty-six blocks has numbers from the four groups.

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

The author has found 262 magic tours by maintaining the even distribution of numbers but keeping the sum of rows and columns of each quad between 282 and 298. Fig. 2 and Fig. 3 have the unique property of diagonals differing by 4 and their sum differing by 8 from twice the magic constant. That is, |OD – ED| = 4 and |OD + ED – 1740| = 8. Fig. 4 and Fig.5 are closer to ‘perfection’ since they have |OD – ED| = 2 and |OD + ED – 1740| = 6.
 ``` 99 10 117 64 107 2 109 72 35 74 37 144 118 63 100 9 70 111 106 3 142 39 34 75 11 98 65 116 1 108 71 110 73 36 143 38 62 119 8 101 112 69 4 105 40 141 76 33 97 12 115 66 103 6 113 68 77 32 41 140 120 61 102 7 114 67 104 5 138 43 30 79 13 96 59 122 23 86 51 130 31 78 139 42 60 121 14 95 52 129 24 85 44 137 80 29 93 16 123 58 87 22 131 50 25 84 45 136 124 57 94 15 128 53 88 21 134 47 28 81 17 92 55 126 19 90 49 132 83 26 135 46 56 125 18 91 54 127 20 89 48 133 82 27 ``` Fig. 2. OD=868 ED=864 ``` 99 10 117 64 107 2 109 72 35 74 37 144 118 63 100 9 70 111 106 3 142 39 34 75 11 98 65 116 1 108 71 110 73 36 143 38 62 119 8 101 112 69 4 105 40 141 76 33 97 12 115 66 103 6 113 68 31 78 41 140 120 61 102 7 114 67 104 5 42 139 32 77 13 96 59 122 23 86 51 130 79 30 137 44 60 121 14 95 52 129 24 85 138 43 80 29 93 16 123 58 87 22 131 50 25 84 45 136 124 57 94 15 128 53 88 21 134 47 28 81 17 92 55 126 19 90 49 132 83 26 135 46 56 125 18 91 54 127 20 89 48 133 82 27 ``` Fig. 3. OD=868 ED=864 ``` 99 10 117 64 107 2 109 72 35 74 37 144 118 63 100 9 70 111 106 3 142 39 34 75 11 98 65 116 1 108 71 110 73 36 143 38 62 119 8 101 112 69 4 105 40 141 76 33 97 12 115 66 103 6 113 68 31 78 41 140 120 61 102 7 114 67 104 5 42 139 32 77 13 96 53 128 19 90 51 130 79 30 137 44 60 121 18 91 52 129 24 85 138 43 80 29 95 14 127 54 89 20 131 50 25 84 45 136 122 59 92 17 126 55 86 23 134 47 28 81 15 94 57 124 21 88 49 132 83 26 135 46 58 123 16 93 56 125 22 87 48 133 82 27 ``` Fig. 4. OD=868 ED=866 ``` 99 10 117 64 107 2 109 72 35 74 37 144 118 63 100 9 70 111 106 3 142 39 34 75 11 98 65 116 1 108 71 110 73 36 143 38 62 119 8 101 112 69 4 105 40 141 76 33 97 12 115 66 103 6 113 68 77 32 41 140 120 61 102 7 114 67 104 5 138 43 30 79 13 96 53 128 19 90 51 130 31 78 139 42 60 121 18 91 52 129 24 85 44 137 80 29 95 14 127 54 89 20 131 50 25 84 45 136 122 59 92 17 126 55 86 23 134 47 28 81 15 94 57 124 21 88 49 132 83 26 135 46 58 123 16 93 56 125 22 87 48 133 82 27 ``` Fig. 5. OD=868 ED=866
[Editor's note: All the magic squares in this article are shown in the same orientation as the original Willcocks tour for purposes of comparison. If the conventions used in the ‘Knight's Tour Notes’ catalogue of 8×8 tours were followed the numbering of the tours would be reversed (to give 1 in the corner instead of 144) and rotated 90 degrees anticlockwise, to place the 1 at the top left corner.]
Figures 6, 7, 8 and 9, also derived from the Willcocks tour, are ‘almost perfect magic tours’ in which the sum of the odd diagonal is equal to the magic constant. [Of course, reversing the numbering gives a tour with even diagonal magic.]
 ``` 99 10 117 64 107 2 109 72 35 74 37 144 118 63 100 9 70 111 106 3 142 39 34 75 11 98 65 116 1 108 71 110 73 36 143 38 62 119 8 101 112 69 4 105 40 141 76 33 97 12 115 66 103 6 113 68 77 32 41 140 120 61 102 7 114 67 104 5 138 43 30 79 13 96 55 126 23 86 53 128 31 78 139 42 60 121 22 87 54 127 24 85 44 137 80 29 95 14 125 56 89 20 129 52 25 84 45 136 122 59 88 21 130 51 90 19 134 47 28 81 15 94 57 124 17 92 49 132 83 26 135 46 58 123 16 93 50 131 18 91 48 133 82 27 ``` Fig. 6. OD=870 ED=862 ``` 99 10 117 64 107 2 109 72 35 74 37 144 118 63 100 9 70 111 106 3 142 39 34 75 11 98 65 116 1 108 71 110 73 36 143 38 62 119 8 101 112 69 4 105 40 141 76 33 97 12 115 66 103 6 113 68 31 78 41 140 120 61 102 7 114 67 104 5 42 139 32 77 13 96 55 126 23 86 53 128 79 30 137 44 60 121 14 95 54 127 24 85 138 43 80 29 93 16 125 56 87 22 129 52 25 84 45 136 122 59 94 15 130 51 88 21 134 47 28 81 17 92 57 124 19 90 49 132 83 26 135 46 58 123 18 91 50 131 20 89 48 133 82 27 ``` Fig. 7. OD=870 ED=866 ``` 99 10 117 64 107 2 109 72 35 74 37 144 118 63 100 9 70 111 106 3 142 39 34 75 11 98 65 116 1 108 71 110 73 36 143 38 62 119 8 101 112 69 4 105 40 141 76 33 97 12 115 66 103 6 113 68 31 78 41 140 120 61 102 7 114 67 104 5 42 139 32 77 13 96 55 126 23 86 53 128 79 30 137 44 60 121 22 87 54 127 24 85 138 43 80 29 95 14 125 56 89 20 129 52 25 84 45 136 122 59 88 21 130 51 90 19 134 47 28 81 15 94 57 124 17 92 49 132 83 26 135 46 58 123 16 93 50 131 18 91 48 133 82 27 ``` Fig. 8. OD=870 ED=862 ``` 99 10 117 64 107 2 109 72 35 74 37 144 118 63 100 9 70 111 106 3 142 39 34 75 11 98 65 116 1 108 71 110 73 36 143 38 62 119 8 101 112 69 4 105 40 141 76 33 97 12 115 66 103 6 113 68 77 32 41 140 120 61 102 7 114 67 104 5 138 43 30 79 13 96 55 126 23 86 53 128 31 78 139 42 60 121 14 95 54 127 24 85 44 137 80 29 93 16 125 56 87 22 129 52 25 84 45 136 122 59 94 15 130 51 88 21 134 47 28 81 17 92 57 124 19 90 49 132 83 26 135 46 58 123 18 91 50 131 20 89 48 133 82 27 ``` Fig. 9. OD=870 ED=866

The author has enumerated sixty-five ‘almost perfect magic tours’ with the even diagonal equal to magic constant and a few of them are shown from Fig.10 to Fig 21. Others can be derived from these tours by keeping the quads containing the even diagonal intact and rearranging the other quads.
 ``` 99 10 117 64 107 2 109 72 35 74 37 144 118 63 100 9 70 111 106 3 142 39 34 75 11 98 65 116 1 108 71 110 73 36 143 38 62 119 8 101 112 69 4 105 40 141 76 33 97 12 115 66 103 6 113 68 77 32 139 42 120 61 102 7 114 67 104 5 140 41 78 31 13 96 55 126 19 90 53 128 29 80 43 138 60 121 18 91 54 127 20 89 44 137 30 79 95 14 125 56 87 52 129 24 133 28 81 46 122 59 92 17 130 21 88 49 84 45 136 27 15 94 57 124 51 86 23 132 25 134 47 82 58 123 16 93 22 131 50 85 48 83 26 135 ``` Fig. 10. OD=1000 ED=870 ``` 99 10 117 64 107 2 109 72 35 74 37 144 118 63 100 9 70 111 106 3 142 39 34 75 11 98 65 116 1 108 71 110 73 36 143 38 62 119 8 101 112 69 4 105 40 141 76 33 97 12 115 66 103 6 113 68 77 32 139 42 120 61 102 7 114 67 104 5 140 41 78 31 13 96 55 126 19 90 53 128 29 80 43 138 60 121 18 91 54 127 20 89 44 137 30 79 95 14 125 56 131 52 87 28 129 46 81 26 122 59 92 17 86 21 130 45 88 27 136 47 15 94 57 124 51 132 23 84 49 134 25 82 58 123 16 93 22 85 50 133 24 83 48 135 ``` Fig. 11. OD=956 ED=870 ``` 99 10 117 64 107 2 109 72 35 74 37 144 118 63 100 9 70 111 106 3 142 39 34 75 11 98 65 116 1 108 71 110 73 36 143 38 62 119 8 101 112 69 4 105 40 141 76 33 97 12 115 66 103 6 113 68 77 32 139 42 120 61 102 7 114 67 104 5 140 41 30 79 13 96 55 126 19 90 53 128 31 78 43 138 60 121 18 91 54 127 20 89 44 137 80 29 95 14 125 56 87 22 129 52 81 28 45 136 122 59 92 17 130 51 88 21 48 133 82 27 15 94 57 124 23 86 49 132 25 84 135 46 58 123 16 93 50 131 24 85 134 47 26 83 ``` Fig. 12. OD=1072 ED=870 ``` 99 10 117 64 107 2 109 72 35 74 37 144 118 63 100 9 70 111 106 3 142 39 34 75 11 98 65 116 1 108 71 110 73 36 143 38 62 119 8 101 112 69 4 105 40 141 76 33 97 12 115 66 103 6 113 68 77 32 139 42 120 61 102 7 114 67 104 5 140 41 30 79 13 96 55 126 19 90 53 128 31 78 43 138 60 121 18 91 54 127 20 89 44 137 80 29 95 14 125 56 87 22 129 52 81 28 45 136 122 59 92 17 130 51 88 21 134 47 82 27 15 94 57 124 23 86 49 132 25 84 135 46 58 123 16 93 50 131 24 85 48 133 26 83 ``` Fig. 13. OD=986 ED=870 ``` 99 10 117 64 107 2 109 72 35 74 37 144 118 63 100 9 70 111 106 3 142 39 34 75 11 98 65 116 1 108 71 110 73 36 143 38 62 119 8 101 112 69 4 105 40 141 76 33 97 12 115 66 103 6 113 68 77 32 41 140 120 61 102 7 114 67 104 5 138 43 30 79 13 96 55 126 19 90 53 128 31 78 139 42 60 121 18 91 54 127 24 85 44 137 80 29 95 14 125 56 89 20 129 52 25 84 45 136 122 59 92 17 130 51 86 23 48 133 28 81 15 94 57 124 21 88 49 132 83 26 135 46 58 123 16 93 50 131 22 87 134 47 82 27 ``` Fig. 14. OD=956 ED=870 ``` 99 10 117 64 107 2 109 72 35 74 37 144 118 63 100 9 70 111 106 3 142 39 34 75 11 98 65 116 1 108 71 110 73 36 143 38 62 119 8 101 112 69 4 105 40 141 76 33 97 12 115 66 103 6 113 68 31 78 41 140 120 61 102 7 114 67 104 5 42 139 32 77 13 96 55 126 15 94 53 128 79 30 137 44 60 121 14 95 54 127 24 85 138 43 80 29 91 18 125 56 93 16 129 52 25 84 45 136 122 59 92 17 130 51 86 23 48 133 28 81 19 90 57 124 21 88 49 132 83 26 135 46 58 123 20 89 50 131 22 87 134 47 82 27 ``` Fig. 15. OD=956 ED=870 ``` 99 10 117 64 107 2 109 72 35 74 37 144 118 63 100 9 70 111 106 3 142 39 34 75 11 98 65 116 1 108 71 110 73 36 143 38 62 119 8 101 112 69 4 105 40 141 76 33 97 12 115 66 103 6 113 68 77 32 139 42 120 61 102 7 114 67 104 5 140 41 78 31 13 96 55 122 19 94 45 132 29 84 43 138 60 121 18 95 54 123 28 85 44 133 30 79 91 14 125 56 93 20 131 46 83 26 137 48 126 59 92 17 124 53 86 27 134 47 80 25 15 90 57 128 21 88 51 130 23 82 49 136 58 127 16 89 52 129 22 87 50 135 24 81 ``` Fig. 16. OD=888 ED=870 ``` 99 10 117 64 107 2 109 72 35 74 37 144 118 63 100 9 70 111 106 3 142 39 34 75 11 98 65 116 1 108 71 110 73 36 143 38 62 119 8 101 112 69 4 105 40 141 76 33 97 12 115 66 103 6 113 68 77 32 139 42 120 61 102 7 114 67 104 5 140 41 78 31 13 96 55 122 19 94 45 136 25 84 43 138 60 121 18 95 54 123 24 85 44 137 30 79 91 14 125 56 93 20 135 46 83 26 133 48 126 59 92 17 124 53 86 23 134 47 80 29 15 90 57 128 21 88 51 130 27 82 49 132 58 127 16 89 52 129 22 87 50 131 28 81 ``` Fig. 17. OD=888 ED=870 ``` 99 10 117 64 107 2 109 72 35 74 37 144 118 63 100 9 70 111 106 3 142 39 34 75 11 98 65 116 1 108 71 110 73 36 143 38 62 119 8 101 112 69 4 105 40 141 76 33 97 12 115 66 103 6 113 68 77 32 139 42 120 61 102 7 114 67 104 5 140 41 78 31 13 96 55 122 19 94 45 136 29 80 43 138 60 121 18 95 54 123 28 81 44 137 30 79 91 14 125 56 93 20 135 46 83 26 133 48 126 59 92 17 124 53 82 27 134 47 84 25 15 90 57 128 21 88 51 130 23 86 49 132 58 127 16 89 52 129 22 87 50 131 24 85 ``` Fig. 18. OD=888 ED=870 ``` 99 10 117 64 107 2 109 72 35 74 37 144 118 63 100 9 70 111 106 3 142 39 34 75 11 98 65 116 1 108 71 110 73 36 143 38 62 119 8 101 112 69 4 105 40 141 76 33 97 12 115 66 103 6 113 68 77 32 139 42 120 61 102 7 114 67 104 5 140 41 78 31 13 96 59 122 15 94 49 136 25 80 43 138 60 121 14 95 58 123 24 81 48 137 30 79 91 18 125 56 93 16 135 50 83 26 133 44 126 55 92 17 124 57 82 23 134 47 84 29 19 90 53 128 21 88 51 130 27 86 45 132 54 127 20 89 52 129 22 87 46 131 28 85 ``` Fig. 19. OD=888 ED=870 ``` 99 10 117 64 107 2 109 72 35 74 37 144 118 63 100 9 70 111 106 3 142 39 34 75 11 98 65 116 1 108 71 110 73 36 143 38 62 119 8 101 112 69 4 105 40 141 76 33 97 12 115 66 103 6 113 68 77 32 139 42 120 61 102 7 114 67 104 5 140 41 78 31 13 96 59 122 19 90 45 132 29 84 43 138 60 121 18 91 58 123 28 85 44 133 30 79 95 14 125 56 89 20 131 46 83 26 137 48 126 55 92 17 124 57 86 27 134 47 80 25 15 94 53 128 21 88 51 130 23 82 49 136 54 127 16 93 52 129 22 87 50 135 24 81 ``` Fig. 20. OD=888 ED=870 ``` 99 10 117 64 107 2 109 72 35 74 37 144 118 63 100 9 70 111 106 3 142 39 34 75 11 98 65 116 1 108 71 110 73 36 143 38 62 119 8 101 112 69 4 105 40 141 76 33 97 12 115 66 103 6 113 68 77 32 139 42 120 61 102 7 114 67 104 5 140 41 78 31 13 96 59 126 15 90 49 136 25 80 43 138 60 121 14 91 58 127 24 81 48 137 30 79 95 18 125 56 89 16 135 50 83 26 133 44 122 55 92 17 128 57 82 23 134 47 84 29 19 94 53 124 21 88 51 130 27 86 45 132 54 123 20 93 52 129 22 87 46 131 28 85 ``` Fig. 21. OD=888 ED=870

 Four Perfect Magic Tours. by A. Kumar é
Figures 22, 23, 24 and 25 are the ‘perfect magic tours’, with both diagonals magic.

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

Dedication: T. H. Willcocks (1968) raised following question in his paper. “Is there a fully magic tour on a 12×12 board? Can any reader find one or a closer approximation than that of Fig.1?” The author has settled this question. Enumeration of magic tours has remained elusive and more effort is required in this direction. (The author dedicates this paper to Mr. T. H. Willcocks whose seminal works on magic tours have attracted, entertained, motivated and enchanted innumerable people. The author is obliged to Mr. G. P. Jelliss for providing photocopies of Willcocks' papers.)

 A New Type of Magic Tour. by A. Kumar é

Figures 26 and 27 are curious magic tours since the sums of their diagonals are equal but different from the magic constant.

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

These are believed to be the only magic tours of the kind so far found on boards of all sizes. Readers are requested to look for more such magic tours.

 Emperor Magic Tours by G. P. Jelliss é
The tours shown in this section are sometimes known as two-knight tours or four-knight tours since they consist of two or four sequences of knight moves joined by rook moves. The unorthodox chess-piece that combines the moves of knight and rook is known in Variant Chess as an empress and when the rook moves are restricted to single step (wazir) moves the piece (knight + wazir) is called an emperor. Since the moves of the emperor always take it to a cell of opposite colour to that on which it stands, the results of the theorems in Issue 25, regarding even-sided boards apply to it. The question of the possibility or not of magic knight's tours on rectangles 4m by 4n + 2, led me to consider emperor tours on such boards. As expected, they are possible. Examples follow. I am working on a new page on the Knight's Tour Notes website with details of previous work on tours of these types on the 8×8 board.
The following two are the only two-knight emperor magic tours on the 4×6 board. The magic constants are 50 and 75. These tours can be renumbered cyclically from the half-way point (i.e. 12-13 becomes 24-1 and vice versa) and remain magic but are then empress tours since the 12-13 rook-move link is a three-cell move.

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

Here are examples 8 by 6. The first was consciously constructed using ‘contiguous contraparallel chains’, but the others were constructed for their visual symmetry. In the first tour if the ranks are divided into three pairs then all these pairs add to 49 (rank total therefore 3×49 = 147. In the files pairs related by reflection in the horizontal median add to 25 or 73 which together equal 98 (twice 49) thus ensuring the file sum is 4×49 = 196. This is the same in the second tour, but in the third tour the constants in the files are 37 and 61.

 ```34 15 46 3 32 17 47 2 33 16 5 44 14 35 4 45 18 31 1 48 13 36 43 6 24 25 12 37 30 19 11 38 21 28 7 42 26 23 40 9 20 29 39 10 27 22 41 8 ``` ```15 32 45 4 17 34 44 3 16 33 46 5 31 14 1 48 35 18 2 43 36 13 6 47 23 30 37 12 19 26 42 11 24 25 38 7 29 22 9 40 27 20 10 41 28 21 8 39 ``` ```42 5 36 13 44 7 35 22 43 6 27 14 4 41 12 37 8 45 21 34 23 26 15 28 40 3 38 11 46 9 33 20 25 24 29 16 2 39 18 31 10 47 19 32 1 48 17 30 ```

Before going on to the 8×10 board, here are some 8×8 examples of similar type.

The following tour was found as an offshoot from a study of maximum number of 3-move knight-lines in tours of squares and diamonds type. The squares and diamonds are joined to make 12 three-unit lines, thus forming four knight-paths. When the ends of the knight paths are joined by wazir moves this gives a magic emperor (wazir + knight) tour. (Not diagonally magic: DO = 288, DE = 240.)

 ``` 6 27 64 33 32 1 38 59 63 34 5 28 37 60 31 2 26 7 36 61 4 29 58 39 35 62 25 8 57 40 3 30 14 19 56 41 24 9 46 51 55 42 13 20 45 52 23 10 18 15 44 53 12 21 50 47 43 54 17 16 49 48 11 22 ```

The next tour was formed after a study of the structure of the first magic knight tour published by William Beverley in 1848. The righthand half of his tour is symmetrical (with horizontal axis). My idea was that perhaps Beverley started from a biaxially symmetric pattern (the H ‘crosspatch’ pattern) and fiddled around with the lefthand side until he found the solution using what we now call ‘beverley quartes’. My tour replaces the four knight moves in the braid on c3d3-c6d6 by four wazir links. The result is a magic emperor tour whose right-hand side is the same as the Beverley tour, both arithmetically and geometrically. (DO = 240, DE = 280, ading to 520, twice the magic constant.)

 ``` 3 30 45 52 5 28 43 54 46 51 4 29 44 53 6 27 31 2 49 48 25 8 55 42 50 47 32 1 56 41 26 7 15 18 33 64 9 24 39 58 34 63 16 17 40 57 10 23 19 14 61 36 21 12 59 38 62 35 20 13 60 37 22 11 ```

There is no proof that Beverley found his tour by this method. Another, equally possible method based on ‘Contiguous Contraparallel Chains’ was described by H. J. R. Murray. Alternatively it may be that Beverley found his tour by some sort of search method, since his is the first of the regular tours to be encountered when they are arranged in sequence according to the Frénicle method for magic squares (i.e. oriented with the smallest number at the top left corner and the number to its right less than the number below it, and similar tours arranged according to the numbers when read row by row).

A similar magic empress (rook + knight) tour is possible by connecting d4-c4, d5-c5, c3-c6, d3-d6. The righthand side of this has the same geometrical form as the Beverley tour, but not the same arithmetical form. (The numbers on the diagonals are the same as for the previous tour, but permuted.)

 ```51 46 29 4 53 44 27 6 30 3 52 45 28 5 54 43 47 50 1 32 41 56 7 26 2 31 48 49 8 25 42 55 63 34 17 16 57 40 23 10 18 15 64 33 24 9 58 39 35 62 13 20 37 60 11 22 14 19 36 61 12 21 38 59 ```

The next diagram shows that the same plan as for the first 8x6 tour above works on the 8×8 board, the right-hand side being geometrically like the Beverley tour, but not arithmetically. The horizontal pairs all add to 65, and the vertical pairs to 97 and 33 which together equal 130. (The diagonal sums are DO = 360, DE = 288).

 ```46 19 62 3 44 21 6 59 63 2 45 20 5 60 43 22 18 47 4 61 24 41 58 7 1 64 17 48 57 8 23 42 32 33 16 49 40 25 10 55 15 50 29 36 9 56 39 26 34 31 52 13 28 37 54 11 51 14 35 30 53 12 27 38 ```

Now we come to the 8×10 example, on the same plan as the previous 8×8 tour: the horizontal pairs all add to 81 and the vertical pairs add to 41 and 121 giving the magic constants: row sum R = 5×81 = 405 and file sum S = 4×81 = 324.

 ```58 23 78 3 26 55 74 7 28 53 79 2 57 24 75 6 27 54 73 8 22 59 4 77 56 25 10 71 52 29 1 80 21 60 5 76 51 30 9 72 40 41 20 61 36 45 70 11 32 49 19 62 37 44 65 16 31 50 69 12 42 39 64 17 46 35 14 67 48 33 63 18 43 38 15 66 47 34 13 68 ```

Evidently this scheme can be extended to any boards 8×2n (with n > 2).

 Quasi-Magic Tours by G. P. Jelliss é
By a quasi-magic tour I mean a rectangular (or square) tour in which the ranks add to the magic constant but the files add to two different values (or vice versa). Thus a quasi-magic tour is a special type of semi-magic tour. The following quasi-magic tours have been discovered in course of my attempts to construct magic tours on rectangular boards of dimensions 4h by 2k where k is odd, or to prove that such magic tours are impossible.
We begin with the smallest such board, namely 4×6. On this I find there are just two quasi-magic knight tours, and one two-knight emperor tour, summing to 75 in the 6-cell lines. The sums in the ranks are (a) 52,52,46,46,52,52, (b) 58,58,46,46,46,46, (c) 38,38,38,62,62,62. (See the previous section for two fully magic emperor examples.)

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

There are no quasi-magic 4×6 knight tours with the 4-cell lines magic. However there are 15 two-knight emperor tours. The 4-cell lines add to 50 (all being composed of pairs adding to 25). The ranks in the first two add to 73 and 77. The ranks in the others all add to 69 and 81 in various sequences.

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

I have constructed nine 8×10 tours (three typical examples shown here) based on extending the braid in the Beverley type tours on the 8×8 to cover the extra two files. They all add to 324 in the files, as required in a magic tour, but the ranks sum to two alternating values. The first has the sums 417 and 393, the second 445 and 365, the third 415 and 395.

 ```22 43 80 17 76 15 74 13 28 49 79 18 21 44 25 46 27 48 73 12 20 23 42 77 16 75 14 71 50 29 41 78 19 24 45 26 47 30 11 72 62 3 40 57 36 55 34 51 70 9 39 58 61 4 65 6 67 10 31 52 60 63 2 37 56 35 54 33 8 69 1 38 59 64 5 66 7 68 53 32 ``` ```22 43 80 17 76 45 26 15 74 47 79 18 21 44 25 16 75 46 27 14 20 23 42 77 50 71 12 29 48 73 41 78 19 24 11 30 49 72 13 28 62 3 40 57 70 51 32 9 68 53 39 58 61 4 31 10 69 52 33 8 60 63 2 37 56 65 6 35 54 67 1 38 59 64 5 36 55 66 7 34 ``` ```22 43 80 17 76 15 74 47 28 13 79 18 21 44 25 46 27 14 49 72 20 23 42 77 16 75 48 73 12 29 41 78 19 24 45 26 11 30 71 50 62 3 40 57 36 55 70 51 10 31 39 58 61 4 65 6 33 8 69 52 60 63 2 37 56 35 54 67 32 9 1 38 59 64 5 66 7 34 53 68 ```

This final example is constructed by the ‘lozenge’ method that I found for 12×12 magic tours, but due to the limitations of this board the result is only quasi-magic. The 10-cell lines add to 405 in five pairs adding to 81. The 8-cell lines add to 364 and 284 (the magic constant would be 324).

 ```65 70 67 60 63 18 21 14 11 16 68 57 64 19 22 59 62 17 24 13 71 66 69 58 61 20 23 12 15 10 56 29 6 73 54 27 8 75 52 25 5 72 55 28 7 74 53 26 9 76 30 35 32 3 38 41 80 49 46 51 33 4 37 42 79 2 39 44 77 48 36 31 34 1 40 43 79 47 50 45 ```

Here are some 12×6 examples constructed by joining together two of Awani Kumar's 6×6 semimagic tours, suitably chosen. [See Knight's Tour Notes: 6×6 Semi-Magic Tours]. The 6-cell lines add to 219 and the 12-cell lines to 510 and 366 (the magic constant would be 438).

 ```62 1 4 69 12 71 26 37 28 45 48 35 3 68 61 72 5 10 39 44 25 36 29 46 60 63 2 11 70 13 24 27 38 47 34 49 67 16 65 56 9 6 43 40 53 32 21 30 64 59 18 7 14 57 52 23 42 19 50 33 17 66 15 58 55 8 41 54 51 22 31 20 44 19 22 51 30 53 56 67 58 15 18 5 21 50 43 54 23 28 69 14 55 6 59 16 42 45 20 29 52 31 66 57 68 17 4 7 49 34 47 38 27 24 13 70 11 2 63 60 46 41 36 25 32 39 10 65 72 61 8 3 35 48 33 40 37 26 71 12 9 64 1 62 ```

Similar tours can be constructed by piling the 6×6 tours on top of one another to any length.

The question remains; are quasi-magic knight tours on boards 4m by (4n + 2) such as the 8×10 and 12×6 the best that can be achieved or are magic knight tours possible on at least some boards of these proportions? To settle this question is the aim of this on-going work.
Sections on this page: (36) In Search of 12×12 Magic Square Tours. (37) Four Perfect Magic Tours 12×12. (38) A New Type of Magic Tour. (39) Emperor Magic Tours. (40) Quasi-Magic Tours. Top
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Copyright 2003 G. P. Jelliss and A. Kumar.