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.
|
Theorem: In a natural-numbered square every satin adds to the same total.
|
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).
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) dffer 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
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 3x3 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!
DIAGRAM NEEDED HERE
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).
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 |
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 5x5 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.
Two of the 7x7 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 9and 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 |
DIAGRAM NEEDED HERE
No step-sidestep magic knight tours are possible on the 9x9 board, since the primality conditions are not met. (EXPLAIN MORE!]
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 |