Magic Torus Tours

This note was published here in August 2009, but had been on hold for some time. The subject received some mention in Variant Chess #58 (October 2008).
Quasi-magic 4×4 tours added November 2012.


Natural Magic

By the standard natural numbering of a rectangular board, r ranks by s files, we mean the numbering obtained by ‘scanning’ the board line by line from left to right and top to bottom, as in writing or reading text, assigning the successive numbers 1, 2, 3, ..., rs to the cells as we encounter them. If we number the files from left to right and the ranks from top to bottom, so that each cell is specified by the ordered pair of coordinates (x, y) then the standard natural numbering assigned to cell (x, y) is the number x + s(y−1).

More generally we can call numberings that are rotations or reflections of this standard numbering natural. Thus there are eight natural numberings of a rectangle, commencing at any of the four corners and proceeding initially along rank or file. Each of these numberings can be specified by a simple formula as above. Other ways of writing have been taken as standard in different cultures. For example Chinese and Japanese are written from the bottom right corner and proceed from bottom to top and right to left, while Arabic writing is from right to left and top to bottom.

In terms of moves a natural numbering proceeds in {0, 1} wazir moves followed by a leap of type {1, s−1)} from the end of a line to the start of the next line; it can thus be regarded as a tour by a two-pattern leaper. If we regard the board as a cylinder, in which the right-hand side of the sheet is curved round to join the left-hand side then the {1, s−1} can be regarded as a {1, 1} fers move on the cylinder, and the natural numbering is then a king tour.

At one time in ancient Greece inscriptions were written ‘boustrophedonally’, that is in the manner of ploughing, with a reversal of direction at the end of each line instead of a jump. This type of numbering is a special type of wazir or rook tour.


Because it is so familiar, the magic properties of the natural numbering are often overlooked, but they are considerable. The ranks and files do not add to a magic constant but other sets do.

Theorem: In a natural-numbered square every satin adds to the same total.
By a satin in a square n by n we mean a set of n cells with one in each rank and one in each file. Thus in a satin in a naturally numbered square we have n numbers of the form xi + n(yi−1) where i takes the values 1, 2, ..., n. Since there is one entry in each file this ensures that each possible value of x occurs exactly once. Similarly each possible value of y occurs exactly once. The sum of the numbers in the satin is Si[xi + n(yi−1)] = Si[xi] + nS[yi−1] = [1 + 2 + ... + n] + n[0 + 1 + .. + (n−1)] = n(n+1)/2 + n[n(n−1)/2] = n(n^2+1)/2, which is the magic constant.

This property of the satins all adding to the same total is unaltered if the ranks or files of the square are permuted, or of the square is reflected in a diagonal (that is, transposed).


Any array with the sums of satins property I call a satinic square. Natural numberings and their permutes are particular cases of satinic squares.

In any satinic square the pairs of numbers at opposite corners of any rectangle add to the same total, since in a satin any pair of entries can be replaced by the entries at the other corners of the same rectangle and still leave a satin, and the new satin still adds to the same total.

Also in any satinic square two parallel lines (ranks or files) differ by the same number in every pair of cells.

Here is a satinic square that is not a permute of a natural numbering:

 1  2  5  6
 3  4  7  8
 9 10 13 14
11 12 15 16

Quasi-Magic Knight Tours of the Four by Four Torus

In an undated note recently rediscovered (November 2012) I made a search for magic knight tours on the 4×4 torus, considering 40 configurations of the first 8 moves, but concluded that such a tour was impossible.

Relaxing the conditions, the best that I found were some quasi-magic tours, adding to 34 in the files and to 30 or 38 in the ranks, three open tours and two closed:

 1  4 15 10    1  4 15 10    1 14  5 10    1  6 13 10    1 14  5 10
 6  9  2 13   16  7  2 13    6  9  2 13   14  9  2  5    6  9  2 13
11 14  5  8    5 14 11  8   11  4 15  8    7  4 11 16   15  4 11  8
16  7 12  3   12  9  6  3   16  7 12  3   12 15  8  3   12  7 16  3

Magic King Tour of the Four by Eight Torus

This tour appears in the Twentieth Century Standard Puzzle Book (1907) edited by A. Cyril Pearson. The book is a collection of problems previously published in the London Evening Standard newspaper. This is problem XXVI "A Magic Oblong" on page 26 of Part 1. The ranks add to 132 and the files to 66.

 1 10 11 29 28 19 18 16
 9  2 30 12 20 27  7 25
24 31  3 21 13  6 26  8
32 23 22  4  5 14 15 17
This item added here December 2012.

Step-Sidestep Magic Tours

The natural numbering can be regarded as formed by a series of wazir moves on a cylinder board, with a fers move interpolated whenever the next wazir move would enter an already used cell. This is an example of a step side-step tour.

The 3×3 diagonally magic square uses three types of move {0, 1}, {1, 1} and {1, 2} within the boundaries of the square. However, if we think of the board as having its top and bottom edges joined (to make a tube) and the left and right edges (the ends of the tube) joined to make a torus, these moves can be interpreted differently. The simplest way is to interpret them as a series of diagonal {1, 1} steps interrupted by {0, 1} steps. (The torus board is also termed a double cylinder, anchor ring, doughnut or pretzel by various writers.)

Under this interpretation the 3×3 magic tour is another example of a step-sidestep tour. The fact that the ranks and files are magic in this type of tour is related to the fact that in a natural numbering of a square the diagonals, formed of {1, 1} moves, are particular examples of satins.

In a tour of step-sidestep type the first type of move, the step, is the main type of move used. The sidestep move is interpolated whenever the next main step would take the touring piece to an already visited cell. The method is effective in generating magic tours on any square boards of an odd number of cells, and various combinations of generative steps can be used.

On a torus of side n the coordinates of the moves should be kept less than n/2 units (a unit being the length of a wazir move, i.e. the distance from centre to centre of two adjacent cells), this is because a move of n units takes the piece all round the board and back to its initial cell. Thus a move of length k is equivalent to a move of length n−k in the opposite direction. For example on the 3×3 board a (0, 1) wazir move to the right is equivalent to a (0, −2) dabbaba move to the left, and a (1, 1) fers move up to the right is equivalent to a (−2, −2) alfil move in the opposite direction, or to a (1, −2) knight move steeply down to the right, or a (−2, 1) knight move shallowly up to the left!

Magic tours of the step-sidestep form can always be made diagonally magic by ‘rotating’ the torus (more accurately, circularly permuting the ranks or files) to bring the middle number (the average, A) to the centre cell, since the other numbers on the diagonals are then, because of the symmetry of the path, complementary pairs (adding to 2A).


Five-Square Magic Torus Tours

On the 5×5 board there are 24 distinct magic tours of this step-sidestep type. (Where ‘distinct’ means that we do not count rotations and reflections of the magic square as different.) Here are diagrams of them all, oriented acording to the Frénicle rule (smallest numbers in the first two cells at top left) and with the middle number, 13, in the centre cell. The magic sum is 5×13 = 65.

Wazir + Fers (King)
 9  2 25 18 11
 3 21 19 12 10
22 20 13  6  4
16 14  7  5 23
15  8  1 24 17
Fers + Dabbaba
 3 16  9 22 15
20  8 21 14  2
 7 25 13  1 19
24 12  5 18  6
11  4 17 10 23 
Fers + Dabbaba
 4 17 10 23 11
18  6 24 12  5 
 7 25 13  1 19 
21 14  2 20  8
15  3 16  9 22
Fers + Alfil
 7  3 24 20 11 
 4 25 16 12  8
21 17 13  9  5
18 14 10  1 22
15  6  2 23 19
Fers + Alfil
 1 18 10 22 14
20  7 24 11  3
 9 21 13  5 17
23 15  2 19  6
12  4 16  8 25
Wazir + Alfil
 6  3 25 17 14
 5 22 19 11  8
24 16 13 10  2
18 15  7  4 21
12  9  1 23 20
Wazir + Alfil
 8  2 21 20 14
 4 23 17 11 10
25 19 13  7  1
16 15  9  3 22
12  6  5 24 18
Dabbaba + Alfil
 5 16  7 23 14
18  9 25 11  2
 6 22 13  4 20
24 15  1 17  8
12  3 19 10 21
Wazir + Knight
 3 14 25  6 17
22  8 19  5 11
16  2 13 24 10
15 21  7 18  4
 9 20  1 12 23
Wazir + Knight
 2 11 25  9 18
21 10 19  3 12
20  4 13 22  6
14 23  7 16  5
 8 17  1 15 24
Wazir + Knight
 4 12 25  8 16
23  6 19  2 15
17  5 13 21  9
11 24  7 20  3
10 18  1 14 22
Dabbaba + Knight
 3 15 22  9 16
21  8 20  2 14
19  1 13 25  7
12 24  6 18  5
10 17  4 11 23
Dabbaba + Knight
 2 14 21  8 20
23 10 17  4 11
19  1 13 25  7
15 22  9 16  3
 6 18  5 12 24
Dabbaba + Knight
 5 12 24  6 18
22  9 16  3 15
19  1 13 25  7
11 23 10 17  4
 8 20  2 14 21
Fers + Knight
 1 14 22 10 18
24  7 20  3 11
17  5 13 21  9
15 23  6 19  2
 8 16  4 12 25
Fers + Knight
 1 15 24  8 17
23  7 16  5 14
20  4 13 22  6
12 21 10 19  3
 9 18  2 11 25
Fers + Knight
 2 18  9 25 11
19 10 21 12  3
 6 22 13  4 20
23 14  5 16  7
15  1 17  8 24
Fers + Knight
 8 1 24 17 15
 5 23 16 14  7
22 20 13  6  4
19 12 10  3 21
11  9  2 25 18
Alfil + Knight
 3 12 21 10 19
24  8 17  1 15
20  4 13 22  6
11 25  9 18  2
 7 16  5 14 23
Alfil + Knight
 5 11 22  8 19
23  9 20  1 12
16  2 13 24 10
14 25  6 17  3
 7 18  4 15 21
Alfil + Knight
 3 17  6 25 14
19  8 22 11  5
10 24 13  2 16
21 15  4 18  7
12  1 20  9 23
Alfil + Knight
10  1 22 18 14
 3 24 20 11  7
21 17 13  9  5
19 15  6  2 23
12  8  4 25 16
Knight
 3 11 24  7 20
25  8 16  4 12
17  5 13 21  9
14 22 10 18  1
 6 19  2 15 23
Knight
 4 15 21  7 18
25  6 17  3 14
16  2 13 24 10
12 23  9 20  1
 8 19  5 11 22

The first eight use only lateral and diagonal moves. The others all use knight moves for one of the steps, while in the last two both the steps are knight moves, so these two are magic knight tours of the torus. These two tours are given in W. S. Andrews Magic Squares and Cubes 1917 (Figures 19 and 20, page 11).

After entering the first 5 numbers, say in the (1, 2) direction, then there is a choice of directions for the knight sidestep. This cannot be forwards (1, 2) or backwards (−1, −2) since these lead to cells already used, also they cannot be the other ‘vertical’ moves (−1, 2), (1, −2) since the first of these does not alter the file and the second does not alter the rank on which the next sequence of (1, 2) moves begins, so if the step is vertical the sidestep must be horizontal. This is a general rule, applicable to other leapers and larger boards. In the 5×5 case the two horizontal sidesteps at right angles to the step, that is (2, −1) and (−2, 1) are also blocked, since they lead to cells already used; however this is not a general rule.


Seven-Square Magic Torus Tours

There are considerably more such tours on the 7×7 board. Here are the 12 tours in which the step and sidestep are the same type of move. There are four by knight {1, 2}, four by camel {1, 3} and four by zebra {2, 3}. Given the orientation of the first step, say (a, b), the side step can be (b, a), (−b, a), (b, −a), (−b, −a) each of which gives a magic tour, whereas the sidesteps (a, −b) and (−a, b) each give a semimagic tour (that is only the ranks or the files have a uniform sum).

Two of the 7×7 knight tours are pandiagonal, that is all 14 diagonals (lines of cells connected by fers moves) add up to the magic constant. The others are semi-pandiagonal, that is 7 diagonals in one direction and one in the other direction) add to the magic constant. The magic sum is 7×25 = 175.

As for the 5×5 case above the tours are cycled so the middle number 25 comes to the centre, and are oriented by the Frénicle rule.

Knight
 4 23 49 19 38  8 34
41 11 30  7 26 45 15
22 48 18 37 14 33  3
10 29  6 25 44 21 40
47 17 36 13 32  2 28
35  5 24 43 20 39  9
16 42 12 31  1 27 46
Knight
10  5 49 37 32 27 15
41 29 24 19 14  2 46
16 11  6 43 38 33 28
47 42 30 25 20  8  3
22 17 12  7 44 39 34
 4 48 36 31 26 21  9
35 23 18 13  1 45 40
Knight
 8  4 49 38 34 23 19
41 30 26 15 11  7 45
18 14  3 48 37 33 22
44 40 29 25 21 10  6
28 17 13  2 47 36 32
 5 43 39 35 24 20  9
31 27 16 12  1 46 42
Knight
 5 27 49 15 37 10 32
41 14 29  2 24 46 19
28 43 16 38 11 33  6
 8 30  3 25 47 20 42
44 17 39 12 34  7 22
31  4 26 48 21 36  9
18 40 13 35  1 23 45
Camel
11  6 43 38 33 28 16
37 32 27 15 10  5 49
21  9  4 48 36 31 26
47 42 30 25 20  8  3
24 19 14  2 46 41 29
 1 45 40 35 23 18 13
34 22 17 12  7 44 39
Camel
 6 16 33 43 11 28 38
49 10 27 37  5 15 32
36  4 21 31 48  9 26
30 47  8 25 42  3 20
24 41  2 19 29 46 14
18 35 45 13 23 40  1
12 22 39  7 17 34 44
Camel
 3 18 33 48 14 22 37
49  8 23 38  4 19 34
39  5 20 35 43  9 24
29 44 10 25 40  6 21
26 41  7 15 30 45 11
16 31 46 12 27 42  1
13 28 36  2 17 32 47
Camel
14  3 48 37 33 22 18
38 34 23 19  8  4 49
20  9  5 43 39 35 24
44 40 29 25 21 10  6
26 15 11  7 45 41 30
 1 46 42 31 27 16 12
32 28 17 13  2 47 36
Zebra
 4 21 31 48  9 26 36
43 11 28 38  6 16 33
40  1 18 35 45 13 23
30 47  8 25 42  3 20
27 37  5 15 32 49 10
17 34 44 12 22 39  7
14 24 41  2 19 29 46
Zebra
 7 26 45 15 41 11 30
36 13 32  2 28 47 17
23 49 19 38  8 34  4
10 29  6 25 44 21 40
46 16 42 12 31  1 27
33  3 22 48 18 37 14
20 39  9 35  5 24 43
Zebra
 2 24 46 19 41 14 29
39 12 34  7 22 44 17
27 49 15 37 10 32  5
 8 30  3 25 47 20 42
45 18 40 13 35  1 23
33  6 28 43 16 38 11
21 35  9 31  4 26 48
Zebra
 5 20 35 43  9 24 39
48 14 22 37  3 18 33
42  1 16 31 46 12 27
29 44 10 25 40  6 21
23 38  4 19 34 49  8
17 32 47 13 28 36  2
11 26 41  7 15 30 45

The geometry of ‘straight lines’ on the torus is distinctly non-euclidean. A diagram can be drawn showing three straight lines formed by knight moves (2, 1), camel moves (−1, 3) and zebra moves (3, −2) starting at any cell and going in three completely different drections, and yet the three lines pass through the same seven ‘points’ (cell centres), each in a different order!

No step-sidestep magic knight tours are possible on the 9×9 board, since the primality conditions are not met.


Eleven-Square Magic Torus Tours

Four magic knight torus tours are again possible on the 11×11 board as shown here.

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

Notes

A generalisation of this method, in which the numbers are entered on the lines in non-numerical order, is described by W. H. Benson and O. Jacoby in New Recreations with Magic Squares (1976) under the title of ‘New Cyclical Method’. However, I believe the basic idea goes back to C. Planck The Theory of Path Nasiks (1905) and probably to A. H. Frost Quarterly Journal of Mathematics (1878).