ç Knight's Tour Notes Index

Semi-Magic Tours of Knight on 4×n Boards

By George Jelliss (August 2003).

4×6 Board

On the smallest tourable board in this class, the 4×6, I find there are just two quasi-magic knight tours, 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. (These were first shown in The Games and Puzzles Journal, issue 26 April 2003).

Forward 1 22 9 18 5 20 8 17 6 21 10 13 23 2 15 12 19 04 16 7 24 3 14 11 3 18 7 22 11 14 6 21 4 13 8 23 17 2 19 10 15 12 20 5 16 1 24 9

Reverse 5 20 7 16 3 24 12 15 4 19 8 17 21 6 13 10 23 2 14 11 22 1 18 9 5 20 9 24 1 16 8 23 6 15 10 13 19 04 21 12 17 2 22 7 18 3 14 11

There are no quasi-magic 4×6 knight tours with the 4-cell lines magic.

4×7 Board

Note added May 2004: It is also possible to form semi-magic tours on rectangles with one even side and one odd side. This seems not to have been studied in detail. Here is an example, from L'Echiquier, October 1929.

13 24 3 18 7 28 9 2 21 14 25 10 17 6 23 12 19 4 15 8 27 20 1 22 11 26 5 16

The tour has rotational symmetry and the files all add to 58.

4×8 Board

The question of whether magic knight's tours are possible on the 4×8 board was considered by H. J. R. Murray: ‘The knight's tour on the half-chessboard’ British Chess Magazine October 1917, but his results are inaccurate. There are 68 tours (Murray found 67) in which the four lines of eight give the total 132, and there are 8 others in which the lines of four all add to 66 (Jelliss using pen and paper, and Stertenbrink using computer, August 2003). I give here only the special cases in the larger class, but all in the smaller class.

Quasi-magic tours 4×8 with the long lines magic (i.e. short lines add to two different values).

Forward 1 01 30 11 24 07 28 13 18 10 23 08 29 12 17 06 27 31 02 21 16 25 04 19 14 22 09 32 03 20 15 26 05 2 03 22 05 28 09 32 15 18 06 25 02 21 16 19 12 31 23 04 27 08 29 10 17 14 26 07 24 01 20 13 30 11 3 03 30 13 24 01 28 15 18 12 23 02 29 14 17 08 27 31 04 21 10 25 06 19 16 22 11 32 05 20 09 26 07 4 03 32 07 20 09 22 13 26 06 19 04 29 16 25 10 23 31 02 17 08 21 12 27 14 18 05 30 01 28 15 24 11 5 03 32 09 26 05 20 13 24 08 29 04 21 10 25 16 19 31 02 27 06 17 14 23 12 28 07 30 01 22 11 18 15

Reverse 11 24 01 30 13 18 07 28 02 31 12 17 08 29 14 19 23 10 25 04 21 16 27 06 32 03 22 09 26 05 20 15 07 26 09 32 13 20 03 22 10 29 06 25 04 23 16 19 27 08 31 12 17 14 21 02 30 11 28 05 24 01 18 15 11 22 01 28 13 24 07 26 02 29 12 23 08 27 14 17 21 10 31 04 19 16 25 06 30 03 20 09 32 05 18 15 07 20 11 24 13 26 01 30 10 23 08 17 04 29 14 27 19 06 21 12 25 16 31 02 22 09 18 05 32 03 28 15 05 26 03 32 11 22 15 18 02 31 06 27 16 19 10 21 25 04 29 12 23 08 17 14 30 01 24 07 28 13 20 09

Notes:
1 has middle link c-e or d-f, file sums 6 of 64 and 2 of 72 (reverse 6 of 68 and 2 of 60), diametral difference 4.
2 has middle link e-g or b-d, file sums 4 of 58 and 4 of 74, diametral difference 8.
3 has middle link a-c or f-h, file sums 6 of 68 and 2 of 60 (reverse 6 of 64 and 2 of 72), diametral difference 4.
4 has middle link c-e or d-f, file sums 4 of 58 and 4 of 74, diametral difference 8.
5 has middle link e-g or b-d, file sums 6 0f 70 and 2 of 54 (reverse 6 0f 62 and 2 of 78).

Near-magic tours 4×8 with the long lines magic (i.e. short lines add to magic value and two others).

Forward 1 01 30 05 26 15 20 11 24 06 27 16 21 04 25 14 19 31 02 29 08 17 12 23 10 28 07 32 03 22 09 18 13 2 03 22 13 28 05 32 11 18 14 25 04 21 12 17 08 31 23 02 27 16 29 06 19 10 26 15 24 01 20 09 30 07 3 03 24 13 28 05 32 09 18 14 27 04 21 12 17 06 31 23 02 25 16 29 08 19 10 26 15 22 01 20 11 30 07 4 03 30 09 20 05 32 11 22 08 27 04 1 10 21 14 17 29 02 25 06 19 16 23 12 26 07 28 01 24 13 18 15 5 03 30 13 24 01 28 11 22 14 17 02 29 12 23 08 27 31 04 19 16 25 06 21 10 18 15 32 05 20 09 26 07 6 03 32 13 20 05 22 11 26 14 17 04 29 12 25 08 23 31 02 19 16 21 06 27 10 18 15 30 01 28 09 24 07 7 03 32 13 20 05 24 09 26 14 17 04 29 12 27 06 23 31 02 19 16 21 08 25 10 18 15 30 01 28 11 22 07 8 03 32 15 20 05 26 09 22 14 19 04 25 16 21 06 27 31 02 17 12 29 08 23 10 18 13 30 01 24 11 28 07

Reverse 05 26 01 30 11 24 15 20 02 31 04 25 16 21 10 23 27 06 17 12 29 08 19 14 32 03 28 07 18 13 22 09 07 18 09 32 13 24 03 26 10 31 06 17 04 27 14 23 19 08 29 12 21 16 25 02 30 11 20 05 28 01 22 15 07 18 11 32 13 22 03 26 10 31 08 17 04 25 14 23 19 06 29 12 21 16 27 02 30 09 20 05 28 01 24 15 07 26 05 32 09 20 15 18 04 31 08 27 14 17 10 21 25 06 29 02 23 12 19 16 30 03 24 13 28 01 22 11 11 22 05 32 09 20 03 30 06 25 10 21 04 31 16 19 23 12 27 08 17 14 29 02 26 07 24 13 28 01 18 15 07 22 11 28 13 20 01 30 10 25 08 21 04 29 16 19 23 06 27 12 17 14 31 02 26 09 24 05 32 03 18 15 07 24 09 28 13 20 01 30 10 27 06 21 04 29 16 19 23 08 25 12 17 14 31 02 26 11 22 05 32 03 18 15 11 24 07 28 13 18 01 30 06 27 12 17 08 29 14 19 23 10 25 04 21 16 31 02 26 05 22 09 32 03 20 15

Notes:
1 has file sums 2 0f 58, 5 of 66, 1 of 82, and middle move c-e type.
2 and 3 have file sums 2 of 64, 4 0f 66, 2 0f 68, middle move c-e.
4 has file sums 2 of 58, 5 0f 66, 1 of 82, and middle move a-c type.
5 has file sums 1 of 58, 6 of 66, 1 of 74, middle move e-g, diametral difference 4.
6 and 7 have file sums 1 of 62, 6 of 66, 1 of 70, middle move e-g.
8 has file sums 1 of 58, 6 of 66, 1 of 74, middle move c-e, diametral difference 4.
Tours 6 and 7 are those diagrammed by Murray (1917) as being the “nearest approaches” to magic on the 4×8.

Semi-magic tours 4×8 with the 8 short lines magic.

Forward 1 01 30 03 20 05 22 09 26 16 19 14 29 12 25 06 23 31 02 17 04 21 08 27 10 18 15 32 13 28 11 24 07 2 01 30 03 20 05 24 11 26 16 19 14 29 12 27 08 23 31 02 17 04 21 06 25 10 18 15 32 13 28 09 22 07 3 01 30 13 20 05 22 09 26 16 19 04 29 12 25 06 23 31 02 17 14 21 08 27 10 18 15 32 03 28 11 24 07 4 01 30 13 20 05 24 11 26 16 19 04 29 12 27 08 23 31 02 17 14 21 06 25 10 18 15 32 03 28 09 22 07 5 01 30 15 20 05 22 09 26 16 19 02 29 12 25 06 23 31 14 17 04 21 08 27 10 18 03 32 13 28 11 24 07 6 01 30 15 20 05 24 11 26 16 19 02 29 12 27 08 23 31 14 17 04 21 06 25 10 18 03 32 13 28 09 22 07 7 03 32 13 20 05 22 09 26 14 17 04 29 12 25 06 23 31 02 19 16 21 08 27 10 18 15 30 01 28 11 24 07 8 03 32 13 20 05 24 11 26 14 17 04 29 12 27 08 23 31 02 19 16 21 04 25 10 18 15 30 01 28 09 22 07

Reverse 07 24 11 28 13 30 03 32 10 27 08 21 04 19 14 17 23 06 25 12 29 16 31 02 26 09 22 05 20 01 18 15 07 22 09 28 13 30 03 32 10 25 06 21 04 19 14 17 23 08 27 12 29 16 31 02 26 11 24 05 20 01 18 15 07 24 11 28 13 20 03 32 10 27 08 21 04 29 14 17 23 06 25 12 19 16 31 02 26 09 22 05 30 01 18 15 07 22 09 28 13 20 03 32 10 25 06 21 04 29 14 17 23 08 27 12 19 16 31 02 26 11 24 05 30 01 18 15 07 24 11 28 13 18 03 32 10 27 08 21 04 31 14 17 23 06 25 12 29 16 19 02 26 09 22 05 20 01 30 15 07 22 09 28 13 18 03 32 10 25 06 21 04 31 14 17 23 08 27 12 29 16 19 02 26 11 24 05 20 01 30 15 07 24 11 28 13 20 01 30 10 27 08 21 04 29 16 19 23 06 25 12 17 14 31 02 26 09 22 05 32 03 18 15 07 22 09 28 13 20 01 30 10 25 06 21 04 29 16 19 23 08 27 12 17 14 31 02 26 11 24 05 32 03 18 15

Notes: 1 - 6 have middle move a-c or f-h, 7 - 8 have middle move e-g or b-d.
5 - 6 are near-magic, the ranks sum to 1 of 128, 2 of 132, 1 of 136.
7 - 8 are quasimagic, the ranks sum to 2 of 130 and 2 of 134.

Semi-magic tours 4×10. This case has not been looked at so far.

Semi-magic tours 4×12. On this board I have only examined tours of squares and diamonds type, seeking for a possible magic tour, and finding 8 semimagic tours. The short lines (files) are all magic adding to 98. Of these eight tours two are quasimagic.

Forward 1 01 46 21 28 05 30 17 42 09 34 13 38 24 27 04 45 20 43 08 31 16 37 10 35 47 02 25 22 29 06 41 18 33 12 39 14 26 23 48 03 44 19 32 07 40 15 36 11 2 01 46 21 28 05 30 17 42 09 36 15 38 24 27 04 45 20 43 08 31 16 39 12 35 47 02 25 22 29 06 41 18 33 10 37 14 26 23 48 03 44 19 32 07 40 13 34 11 3 01 46 21 28 05 44 17 32 09 34 13 38 24 27 04 45 20 29 08 41 16 37 10 35 47 02 25 22 43 06 31 18 33 12 39 14 26 23 48 03 30 19 42 07 40 15 36 11 4 01 46 21 28 05 44 17 32 09 36 15 38 24 27 04 45 20 29 08 41 16 39 12 35 47 02 25 22 43 06 31 18 33 10 37 14 26 23 48 03 30 19 42 07 40 13 34 11 5 03 48 21 28 05 30 17 42 09 34 13 38 22 25 04 45 20 43 08 31 16 37 10 35 47 02 27 24 29 06 41 18 33 12 39 14 26 23 46 01 44 19 32 07 40 15 36 11 6 03 48 21 28 05 30 17 42 09 36 15 38 22 25 04 45 20 43 08 31 16 39 12 35 47 02 27 24 29 06 41 18 33 10 37 14 26 23 46 01 44 19 32 07 40 13 34 11 7 03 48 21 28 05 44 17 32 09 34 13 38 22 25 04 45 20 29 08 41 16 37 10 35 47 02 27 24 43 06 31 18 33 12 39 14 26 23 46 01 30 19 42 07 40 15 36 11 8 03 48 21 28 05 44 17 32 09 36 15 38 22 25 04 45 20 29 08 41 16 39 12 35 47 02 27 24 43 06 31 18 33 10 37 14 26 23 46 01 30 19 42 07 40 13 34 11

Reverse 11 36 15 40 07 32 19 44 21 28 03 48 14 39 12 33 18 41 06 29 04 45 22 25 35 10 37 16 31 08 43 20 27 24 47 02 38 13 34 09 42 17 30 05 46 01 26 23 11 34 13 40 07 32 19 44 21 28 03 48 14 37 10 33 18 41 06 29 04 45 22 25 35 12 39 16 31 08 43 20 27 24 47 02 38 15 36 09 42 17 30 05 46 01 26 23 11 36 15 40 17 32 05 44 21 28 03 48 14 39 12 33 08 41 20 29 04 45 22 25 35 10 37 16 31 18 43 06 27 24 47 02 38 13 34 09 42 07 30 19 46 01 26 23 11 34 13 40 17 32 05 44 21 28 03 48 14 37 10 33 08 41 20 29 04 45 22 25 35 12 39 16 31 18 43 06 27 24 47 02 38 15 36 09 42 07 30 19 46 01 26 23 11 36 15 40 07 32 19 44 21 28 01 46 14 39 12 33 18 41 06 29 04 45 24 27 35 10 37 16 31 08 43 20 25 22 47 02 38 13 34 09 42 17 30 05 48 03 26 23 11 34 13 40 07 32 19 44 21 28 01 46 14 37 10 33 18 41 06 29 04 45 24 27 35 12 39 16 31 08 43 20 25 22 47 02 38 15 36 09 42 17 30 05 48 03 26 23 11 36 15 40 17 32 05 44 21 28 01 46 14 39 12 33 08 41 20 29 04 45 24 27 35 10 37 16 31 18 43 06 25 22 47 02 38 13 34 09 42 07 30 19 48 03 26 23 11 34 13 40 17 32 05 44 21 28 01 46 14 37 10 33 08 41 20 29 04 45 24 27 35 12 39 16 31 18 43 06 25 22 47 02 38 15 36 09 42 07 30 19 48 03 26 23

Notes:
In 1 and 2 the long lines add to 284, 288, 300, 304.
In 3, 4, 5, 6 the long lines add to 288, 292, 296, 300
Tours 7 and 8 are quasimagic, long lines sum to 292 twice and 296 twice.
1-4 have middle link a-c or j-l. 5-8 have it b-d or i-k.

Addendum: Jean-Charles Meyrignac reports (26 August 2003): My program computed all 4×12 quasi-magic tours. Horizontal lines have their sum equal to 294, and vertical lines have sums alternating between 2 values. Here are the full results [48 tours: they are oriented with the number 1 in the first half of the top rank]:

1 46 15 36 11 44 17 26 9 42 19 28 14 35 12 45 16 25 10 43 18 27 8 41 47 2 33 24 37 4 31 22 39 6 29 20 34 13 48 3 32 23 38 5 30 21 40 7 ----------------------------------------------- 96 96 108 108 96 96 96 96 96 96 96 96 46 1 42 21 40 7 26 17 36 15 32 11 43 22 45 4 25 18 39 8 29 12 35 14 2 47 24 41 20 27 6 37 16 33 10 31 23 44 3 48 5 38 19 28 9 30 13 34 ----------------------------------------------- 114 114 114 114 90 90 90 90 90 90 90 90 46 1 42 21 40 5 28 17 36 15 32 11 43 22 45 4 25 20 37 8 29 12 35 14 2 47 24 41 6 39 18 27 16 33 10 31 23 44 3 48 19 26 7 38 9 30 13 34 ----------------------------------------------- 114 114 114 114 90 90 90 90 90 90 90 90 46 1 42 7 40 11 36 15 32 17 26 21 43 4 45 12 35 14 39 10 25 20 29 18 2 47 6 41 8 37 24 33 16 31 22 27 5 44 3 48 13 34 9 38 23 28 19 30 ----------------------------------------------- 96 96 96 108 96 96 108 96 96 96 96 96 26 1 28 23 32 9 44 21 36 15 42 17 29 4 25 8 47 22 35 10 43 18 39 14 2 27 6 31 24 33 20 45 12 37 16 41 5 30 3 48 7 46 11 34 19 40 13 38 ----------------------------------------------- 62 62 62 110 110 110 110 110 110 110 110 110 30 1 26 21 46 5 36 19 44 15 40 11 27 22 29 6 25 20 45 16 35 12 43 14 2 31 24 47 4 33 8 37 18 41 10 39 23 28 3 32 7 48 17 34 9 38 13 42 ----------------------------------------------- 82 82 82 106 82 106 106 106 106 106 106 106 5 28 1 32 23 46 11 34 19 42 15 38 2 25 4 47 8 33 20 45 12 39 18 41 29 6 27 24 31 22 35 10 43 16 37 14 26 3 30 7 48 9 44 21 36 13 40 17 ----------------------------------------------- 62 62 62 110 110 110 110 110 110 110 110 110 5 42 1 38 9 34 23 48 19 32 15 28 2 39 4 35 24 37 20 33 22 29 18 31 43 6 41 10 45 8 25 12 47 16 27 14 40 3 44 7 36 11 46 21 26 13 30 17 ----------------------------------------------- 90 90 90 90 114 90 114 114 114 90 90 90 23 26 1 32 5 48 17 36 9 40 15 42 2 29 22 25 20 33 8 45 16 43 12 39 27 24 31 4 47 6 35 18 37 10 41 14 30 3 28 21 34 19 46 7 44 13 38 11 ----------------------------------------------- 82 82 82 82 106 106 106 106 106 106 106 106 23 26 1 32 5 46 19 36 9 40 15 42 2 29 22 25 20 35 6 45 16 43 12 39 27 24 31 4 33 18 47 8 37 10 41 14 30 3 28 21 48 7 34 17 44 13 38 11 ----------------------------------------------- 82 82 82 82 106 106 106 106 106 106 106 106 21 28 1 32 19 26 17 44 7 40 13 46 2 33 20 27 24 37 6 39 12 45 8 41 29 22 35 4 31 18 25 16 43 10 47 14 34 3 30 23 36 5 38 11 48 15 42 9 ----------------------------------------------- 86 86 86 86 110 86 86 110 110 110 110 110 23 26 1 32 19 34 7 46 9 40 15 42 2 29 22 25 6 47 18 35 16 43 12 39 27 24 31 4 33 20 45 8 37 10 41 14 30 3 28 21 48 5 36 17 44 13 38 11 ----------------------------------------------- 82 82 82 82 106 106 106 106 106 106 106 106 15 34 1 44 17 36 11 42 19 26 9 40 2 45 16 35 12 43 18 25 10 41 20 27 33 14 47 4 31 24 37 6 29 22 39 8 46 3 32 13 48 5 30 23 38 7 28 21 ----------------------------------------------- 96 96 96 96 108 108 96 96 96 96 96 96 15 36 1 46 17 26 11 44 19 28 9 42 2 47 16 25 12 45 18 27 10 43 20 29 35 14 37 4 33 24 39 6 31 22 41 8 48 3 34 13 38 5 32 23 40 7 30 21 ----------------------------------------------- 100 100 88 88 100 100 100 100 100 100 100 100 44 19 46 1 42 3 38 15 30 9 36 11 47 22 43 18 25 16 29 4 37 12 33 8 20 45 24 41 2 27 14 39 6 31 10 35 23 48 21 26 17 40 5 28 13 34 7 32 ----------------------------------------------- 134 134 134 86 86 86 86 86 86 86 86 86 44 7 42 1 38 11 36 21 34 17 28 15 41 4 45 8 47 22 25 12 27 14 31 18 6 43 2 39 24 37 10 35 20 33 16 29 3 40 5 46 9 48 23 26 13 30 19 32 ----------------------------------------------- 94 94 94 94 118 118 94 94 94 94 94 94 30 7 28 1 48 9 36 21 44 17 38 15 27 4 31 8 33 24 45 12 37 14 41 18 6 29 2 25 10 47 22 35 20 43 16 39 3 26 5 32 23 34 11 46 13 40 19 42 ----------------------------------------------- 66 66 66 66 114 114 114 114 114 114 114 114 26 23 40 1 36 21 42 3 34 19 44 5 39 12 25 22 41 2 35 20 43 4 33 18 24 27 10 37 14 29 8 47 16 31 6 45 11 38 13 28 9 48 15 30 7 46 17 32 ----------------------------------------------- 100 100 88 88 100 100 100 100 100 100 100 100 26 5 46 1 30 23 42 11 40 19 36 15 47 2 25 8 43 10 31 22 33 16 39 18 6 27 4 45 24 29 12 41 20 37 14 35 3 48 7 28 9 44 21 32 13 34 17 38 ----------------------------------------------- 82 82 82 82 106 106 106 106 106 106 106 106 42 5 44 1 34 15 38 11 36 21 28 19 45 2 41 8 39 12 35 16 31 18 25 22 6 43 4 47 14 33 10 37 24 27 20 29 3 46 7 40 9 48 13 32 17 30 23 26 ----------------------------------------------- 96 96 96 96 96 108 96 96 108 96 96 96 44 5 46 1 36 15 40 11 26 21 30 19 47 2 43 8 41 12 25 16 33 18 27 22 6 45 4 37 14 35 10 39 24 29 20 31 3 48 7 42 9 38 13 34 17 32 23 28 ----------------------------------------------- 100 100 100 88 100 100 88 100 100 100 100 100 44 5 40 1 38 11 36 21 34 19 30 15 41 2 43 8 47 22 25 12 27 16 33 18 6 45 4 39 24 37 10 35 20 31 14 29 3 42 7 46 9 48 23 26 13 28 17 32 ----------------------------------------------- 94 94 94 94 118 118 94 94 94 94 94 94 26 5 46 1 44 11 30 21 40 19 36 15 47 2 25 8 29 22 43 12 33 16 39 18 6 27 4 45 24 31 10 41 20 37 14 35 3 48 7 28 9 42 23 32 13 34 17 38 ----------------------------------------------- 82 82 82 82 106 106 106 106 106 106 106 106 30 5 26 1 48 9 36 21 44 19 40 15 27 2 29 8 33 24 45 12 37 16 43 18 6 31 4 25 10 47 22 35 20 41 14 39 3 28 7 32 23 34 11 46 13 38 17 42 ----------------------------------------------- 66 66 66 66 114 114 114 114 114 114 114 114 40 7 34 1 38 11 44 13 26 19 46 15 35 2 39 6 33 24 31 18 45 14 27 20 8 41 4 37 10 43 12 25 22 29 16 47 3 36 9 42 5 32 23 30 17 48 21 28 ----------------------------------------------- 86 86 86 86 86 110 110 86 110 110 110 110 19 42 21 48 1 40 13 28 5 32 11 34 22 45 18 41 16 25 4 37 12 35 8 31 43 20 47 24 39 2 27 14 29 6 33 10 46 23 44 17 26 15 38 3 36 9 30 7 ----------------------------------------------- 130 130 130 130 82 82 82 82 82 82 82 82 19 46 21 28 1 44 13 32 5 36 11 38 22 25 18 45 16 29 4 41 12 39 8 35 47 20 27 24 43 2 31 14 33 6 37 10 26 23 48 17 30 15 42 3 40 9 34 7 ----------------------------------------------- 114 114 114 114 90 90 90 90 90 90 90 90 19 44 23 48 1 40 13 28 5 30 9 34 22 47 20 41 16 25 4 37 12 33 6 31 43 18 45 24 39 2 27 14 29 8 35 10 46 21 42 17 26 15 38 3 36 11 32 7 ----------------------------------------------- 130 130 130 130 82 82 82 82 82 82 82 82 7 30 3 26 1 48 11 46 21 44 17 40 4 27 6 33 10 35 24 37 14 41 20 43 31 8 29 2 25 12 47 22 45 18 39 16 28 5 32 9 34 23 36 13 38 15 42 19 ----------------------------------------------- 70 70 70 70 70 118 118 118 118 118 118 118 23 46 19 42 1 40 5 28 13 36 9 32 20 43 22 25 18 27 14 39 6 33 12 35 47 24 45 2 41 16 29 4 37 10 31 8 44 21 48 17 26 3 38 15 30 7 34 11 ----------------------------------------------- 134 134 134 86 86 86 86 86 86 86 86 86 21 34 17 30 1 26 15 40 11 48 7 44 18 31 20 27 16 29 12 25 14 45 10 47 35 22 33 2 37 24 41 4 39 8 43 6 32 19 36 23 28 3 38 13 42 5 46 9 ----------------------------------------------- 106 106 106 82 82 82 106 82 106 106 106 106 7 32 5 26 1 48 11 46 21 42 15 40 4 29 8 33 10 35 24 37 14 39 18 43 31 6 27 2 25 12 47 22 45 20 41 16 28 3 30 9 34 23 36 13 38 17 44 19 ----------------------------------------------- 70 70 70 70 70 118 118 118 118 118 118 118 19 28 21 34 1 36 15 38 5 42 11 44 22 31 18 27 16 25 2 47 12 45 8 41 29 20 33 24 35 14 37 4 39 6 43 10 32 23 30 17 26 3 48 13 46 9 40 7 ----------------------------------------------- 102 102 102 102 78 78 102 102 102 102 102 102 19 46 21 28 1 42 15 32 5 36 11 38 22 25 18 45 16 31 2 41 12 39 8 35 47 20 27 24 29 14 43 4 33 6 37 10 26 23 48 17 44 3 30 13 40 9 34 7 ----------------------------------------------- 114 114 114 114 90 90 90 90 90 90 90 90 19 30 23 34 1 36 15 38 5 40 9 44 22 33 20 27 16 25 2 47 12 43 6 41 29 18 31 24 35 14 37 4 39 8 45 10 32 21 28 17 26 3 48 13 46 11 42 7 ----------------------------------------------- 102 102 102 102 78 78 102 102 102 102 102 102 3 44 17 34 1 42 19 36 11 40 21 26 16 33 2 43 18 35 12 41 20 25 10 39 45 4 31 14 47 6 29 24 37 8 27 22 32 15 46 5 30 13 48 7 28 23 38 9 ----------------------------------------------- 96 96 96 96 96 96 108 108 96 96 96 96 3 46 17 36 1 44 19 26 11 42 21 28 16 35 2 45 18 25 12 43 20 27 10 41 47 4 33 14 37 6 31 24 39 8 29 22 34 15 48 5 32 13 38 7 30 23 40 9 ----------------------------------------------- 100 100 100 100 88 88 100 100 100 100 100 100 42 19 46 23 48 1 38 3 28 5 32 9 45 22 43 16 39 14 25 12 35 8 29 6 18 41 20 47 24 37 2 27 4 31 10 33 21 44 17 40 15 26 13 36 11 34 7 30 ----------------------------------------------- 126 126 126 126 126 78 78 78 78 78 78 78 42 17 44 23 48 1 38 3 28 7 34 9 45 20 41 16 39 14 25 12 35 10 31 6 18 43 22 47 24 37 2 27 4 29 8 33 21 46 19 40 15 26 13 36 11 32 5 30 ----------------------------------------------- 126 126 126 126 126 78 78 78 78 78 78 78 42 9 40 3 36 1 46 23 32 19 26 17 39 6 43 10 45 12 35 14 25 16 29 20 8 41 4 37 2 47 24 33 22 31 18 27 5 38 7 44 11 34 13 48 15 28 21 30 ----------------------------------------------- 94 94 94 94 94 94 118 118 94 94 94 94 46 9 44 3 40 1 26 23 36 19 30 17 43 6 47 10 25 12 39 14 29 16 33 20 8 45 4 41 2 27 24 37 22 35 18 31 5 42 7 48 11 38 13 28 15 32 21 34 ----------------------------------------------- 102 102 102 102 78 78 102 102 102 102 102 102 42 7 38 3 36 1 46 23 32 21 28 17 39 4 41 10 45 12 35 14 25 18 31 20 8 43 6 37 2 47 24 33 22 29 16 27 5 40 9 44 11 34 13 48 15 26 19 30 ----------------------------------------------- 94 94 94 94 94 94 118 118 94 94 94 94 46 7 42 3 40 1 26 23 36 21 32 17 43 4 45 10 25 12 39 14 29 18 35 20 8 47 6 41 2 27 24 37 22 33 16 31 5 44 9 48 11 38 13 28 15 30 19 34 ----------------------------------------------- 102 102 102 102 78 78 102 102 102 102 102 102 26 23 48 17 44 1 32 13 40 9 34 7 47 20 27 24 29 16 41 4 33 6 37 10 22 25 18 45 2 43 14 31 12 39 8 35 19 46 21 28 15 30 3 42 5 36 11 38 ----------------------------------------------- 114 114 114 114 90 90 90 90 90 90 90 90 40 11 26 21 42 1 36 19 44 3 34 17 27 22 41 12 25 20 43 2 35 18 45 4 10 39 24 29 8 37 14 31 6 47 16 33 23 28 9 38 13 30 7 48 15 32 5 46 ----------------------------------------------- 100 100 100 100 88 88 100 100 100 100 100 100 38 11 36 21 40 1 34 19 42 3 32 17 25 22 39 12 35 20 41 2 33 18 43 4 10 37 24 27 8 47 14 29 6 45 16 31 23 26 9 48 13 28 7 46 15 30 5 44 ----------------------------------------------- 96 96 108 108 96 96 96 96 96 96 96 96 26 21 46 17 42 1 32 15 40 11 36 7 47 18 25 2 45 16 41 12 31 8 39 10 22 27 20 43 24 29 4 33 14 37 6 35 19 48 23 28 3 44 13 30 5 34 9 38 ----------------------------------------------- 114 114 114 90 114 90 90 90 90 90 90 90 46 3 42 9 40 1 36 17 32 19 26 23 43 6 45 2 35 16 39 12 25 22 29 20 4 47 8 41 10 37 14 33 18 31 24 27 7 44 5 48 15 34 11 38 13 28 21 30 ----------------------------------------------- 100 100 100 100 100 88 100 100 88 100 100 100

These 48 of course occur in pairs that are reversals of each other. For example, 14 is the reversal of 1. Also they include the four (including reversals) of squares and diamonds type in the previous list.

Semi-magic tours 4×16 and longer.

The first of the 4×12 tours shown in Jean-Charles Meyrignac's list above, when drawn geometrically, can be seen to be a braid extension of the first tour in the 4×8 section. The method can be extended to the 4×16 board, and to any larger board 4×4k. The result is a quasi-magic tour magic in the long lines and with most of the short lines adding to 32k, and just two of these lines, the third and fourth, adding to 36k.