Semi-Magic Knight's Tours

Part 1: 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.


4×8 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.


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