Links to related pages: Introduction — Part 1: 4 by N Boards — Part 2: 6 by N Boards
8×10 Board Some of the results given here appeared recently in The Games and Puzzles Journal, issue 26, April 2003, where some geometrical diagrams of the tours also appear.
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 365 and 445, the second 395 and 415, the third 393 and 417.
1 38 59 64 5 36 55 66 7 34 60 63 2 37 56 65 6 35 54 67 39 58 61 4 31 10 69 52 33 8 62 3 40 57 70 51 32 9 68 53 41 78 19 24 11 30 49 72 13 28 20 23 42 77 50 71 12 29 48 73 79 18 21 44 25 16 75 46 27 14 22 43 80 17 76 45 26 15 74 47 |
1 38 59 64 5 66 7 34 53 68 60 63 2 37 56 35 54 67 32 9 39 58 61 4 65 6 33 8 69 52 62 3 40 57 36 55 70 51 10 31 41 78 19 24 45 26 11 30 71 50 20 23 42 77 16 75 48 73 12 29 79 18 21 44 25 46 27 14 49 72 22 43 80 17 76 15 74 47 28 13 |
1 38 59 64 5 66 7 68 53 32 60 63 2 37 56 35 54 33 8 69 39 58 61 4 65 6 67 10 31 52 62 3 40 57 36 55 34 51 70 9 41 78 19 24 45 26 47 30 11 72 20 23 42 77 16 75 14 71 50 29 79 18 21 44 25 46 27 48 73 12 22 43 80 17 76 15 74 13 28 49 |
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 78 47 50 45 |
The fourth example above 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 (consisting mainly of pairs adding to 81). The 8-cell lines add to 364 and 284 (the magic constant would be 324).
8×14 Board No results to hand.
8×18 Board These two new examples were constructed by splitting the 8×8 magic tour 00b in two and joining up the loose ends by four paths in direct qaternary symmetry.
42 31 110 107 26 47 128 87 24 85 22 53 134 83 74 7 66 143 111 108 43 30 129 88 25 48 127 54 133 84 21 52 65 142 75 6 32 41 106 109 46 27 130 55 86 23 126 51 62 135 8 73 144 67 105 112 29 44 117 100 89 14 49 132 95 20 9 64 141 68 5 76 40 33 116 101 28 45 56 131 96 13 50 125 136 81 72 77 140 1 113 104 39 36 99 118 15 90 59 122 19 94 63 10 137 4 69 78 34 37 102 115 16 57 120 97 18 91 12 61 124 93 80 71 2 139 103 114 35 38 119 98 17 58 121 60 123 92 11 62 3 138 79 70 |
42 31 110 107 26 53 98 123 126 55 58 11 128 83 74 7 66 143 111 108 43 30 99 124 25 54 97 12 127 84 57 10 65 142 75 6 32 41 106 109 52 27 96 125 122 59 56 13 82 129 8 73 144 67 105 112 29 44 117 100 51 24 95 14 85 130 9 64 141 68 5 76 40 33 116 101 28 45 94 121 50 131 60 15 136 81 72 77 140 1 113 104 39 36 93 118 49 20 23 86 89 132 63 16 137 4 69 78 34 37 102 115 46 21 120 91 48 133 18 61 88 135 80 71 2 139 103 114 35 38 119 92 47 22 19 90 87 134 17 62 3 138 79 70 |
10×10 Board.
Here is Tom Marlow's 10×10 semi-magic tour with quaternary symmetry (i.e. 90 degree rotational symmetry) which appeared in The Games and Puzzles Journal, issue 25, March 2003, where a geometrical diagram is also given. The yellow coloured squares are those containing numbers of the forms 4n + 2 or 4n + 3: there must be an even number of these in a line for it to be magic.
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Links to related pages: Introduction — Part 1: 4 by N Boards — Part 2: 6 by N Boards