All magic knight's tours of the normal chessboard are listed on the linked pages.
Thanks to the most recent work reported below it is believed that all magic knight's tours on the 8×8 board have now been found. How many tours there are depends of course on how they are counted! If they are regarded as geometrical paths on which magic numbering can be imposed then there are 108 tours in all. However some of these are closed paths that can be magically numbered in two or more ways. If we count only open paths then the total becomes 140. Each of these open paths can be numbered from either end, so on this basis the total of arithmetical magic tours is 280. Finally, each of these arithmetical tours can be presented in eight different orientations by rotation and reflection, making a total of 8×280 = 2240 magic arithmetical arrays.
An ‘Excel’ (spreadsheet) version of the catalogue provided by Tim Roberts (but not yet updated with the seven most recent arithmetical tours), can be downloaded by clicking here: mkt8x8.xls. It lists 133 of the 140 tours in arithmetical form and includes diagonal sums. File size 108kB. When you click on the file name, your browser should display it in a new window. Choose ‘Save As’ from the ‘File’ menu to save it in an appropriate folder.
The catalogue derives from successive work by Parmentier (1890s), Lehmann (1932), Murray (1951) and Jelliss (1986) in collecting and classifying the tours. To these have been added the new tours found by T. W. Marlow (1988), by T. S. Roberts (January 2003) and by H. Mackay, J-C. Meyrignac and G. Stertenbrink (June-August 2003).
The catalogue is now divided into three parts.
Part 1 - Rhombic lists the 34 tours formed purely on the ‘squares and diamonds’ plan:
00a, 00b, 00c, 00i, 01d, 01h, 03a, 03b, 05b, 05c, 05f, 05g, 12a, 12b, 12m, 12n, 12o, 12p, 14b, 14d,
23a, 23c, 23d, 23g, 23l, 23m, 23n, 23q, 27i, 27l, 27p, 34d, 34e, 34f.
Part 2 - Beverley lists the 44 tours formed of squares, diamonds and beverley
quartes: 00m, 00d, 00e, 00f, 00g, 00h, 01a, 01b, 01c, 01f, 01g, 03c, 03d, 03e, 03f, 03g,
05d, 05e, 12e, 14a, 14c, 23b, 23e, 23f, 23h, 23i, 23j, 23k, 25a, 25b, 27a, 27b, 27c, 27d,
27j, 27k, 27m, 27q, 27r, 27s, 34a, 34b, 34c, 34g.
Part 3 - Irregular lists the 30 other magic tours: 00j, 00k, 00l, 00n, 00o, 01e, 01i,
05a, 07a, 12c, 12d, 12f, 12g, 12h, 12i, 12j, 12k, 12l, 14e, 14f, 16a, 23o, 23p, 27e, 27f, 27g, 27h,
27n, 27o, 27t.
Since any magic tour can be reflected and rotated (keeping the numbers the right way up!) it can occur in 8 different orientations and only one of these is shown in the catalogue. No 8×8 open knight tour can form a symmetric path, since the middle move 32–33 would have to pass through the centre of symmetry or else cross the axis of symmetry at right angles, neither of which is possible; thus the 8 orientations are always all distinct.
The orientation shown in the catalogue is the one that has the smallest number at the top left corner, and, subject to this condition, the smallest number in the second cell of the first row. Many other schemes of arrangement have been tried, all of which have their advantages and disadvantages. The main advantage of this scheme is that it is applicable to any square array of distinct numbers, and it is the scheme generally accepted, since the publication of the Frénicle/de la Hire 1693 list of 4×4 magic squares, for the arrangement of any such squares.
I am thinking of replacing this system by a slight modification. In this new system the number at the top left will be the smallest ODD corner number. This would result in the odd diagonal always being the principal diagonal (from top left to bottom right).
In each Part of the catalogue, the oriented tours are listed in ascending sequence of the numbers in the successive cells, when read row by row. For example, the first two tours shown in the ‘irregular’ list have all numbers in the first four rows the same, and then differ in the first number of the fifth row, one having 9 and the other 39, so we put the one with the smallest number, 9, first. The close similarity possible means that considerable care is necessary to identify some tours accurately.
I have, at least temporarily, retained the coding introduced in my Chessics catalogue of 1986. In this a two-figure numerical code is assigned to a tour. The code 00 indicates a cyclic tour. Otherwise, the code rs indicates a tour in which the end-points (the cells numbered 1 and 64) are an {r,s} move apart. This coding method enables one to narrow down a given tour to a smallish group. Within the group the various tours are distinguished by small letters a, b, etc.
Any magic tour remains magic when numbered in reverse order. Geometrically these two magic tours are the same path numbered from opposite ends. The numbering that occurs first in our list we call the forward case and the numbering that would occur later we call the reverse case. The tours are shown in this catalogue in both arithmetical and geometrical forms (some more detailed geometrical diagrams are shown in the page on History of Magic Tours). The reverse arithmetical tours are now shown to the right of the tour diagram.
Each reentrant tour (where the cell numbered 64 is a knight's move from the cell numbered 1) determines an endless geometrical path called a closed tour by joining 64 to 1. Some of these closed tours are symmetric by 180° rotation. This corresponds to the numerical property that the numbers in diametrally opposite cells differ by 32. Symmetric tours are printed in Red type. There are 16 symmetric magic tours: 00m and 12a to 12o.
Some closed geometrical magic tours can be numbered from more than one cell to give two or more different arithmetical magic tours. We call such tours cyclic magic tours. The arithmetical tours derived from the same geometrical form can have quite different numerical properties; for example in one numbering the diagonal sum may be maximum, while in another it may be minimum. Although a symmetric closed tour can be renumbered from 33 to 32 and remain magic this does not make it a cyclic tour since the resulting tour is not different from the original, merely a rotation of it.
Cyclic tours are printed in Blue type, and the various different arithmetical forms are distinguished by a capital letter, A, B, etc; and their reversals AR, BR, etc. There is one known 5-fold cyclic tour 00a (lettered A-E), seven 4-fold cyclic 00bcdefgh (lettered A-D), and seven 2-fold cyclic 00ijklmno(lettered A-B). One in this last group 00m is symmetric; it is printed in a Mauve colour, i.e. a combination of red and blue.
In the case of a cyclic magic tour, renumbering x + 1 as 1, x + 2 as 2 and so on will increase a certain number i of entries in a particular rank or file by 64 - x and decrease the other 8 - i entries by x. Thus for the line to remain magic we must have (8-i)x = i(64 -x) which implies i = 8x/64; which shows that i is the same for all ranks and files and that x must be a multiple of 8.
Renumbering a cyclic tour from 8y+1 will have the effect of subtracting 8y from the entries 8y + 1 up to 64, and adding 64 - 8y to the entries 1 up to 8y. Thus the effect on a set of eight entries (such as those in a diagonal) that contains w numbers in the lower range 1 to 8y will be to subtract 8y from all of them and to add 64 to w of them, so their sum will alter by 64w - 8(8y) = 64(w - y), that is a multiple of 64. For example: renumbering 00b from 17, that is y = 2, increases the even diagonal, which has w = 4, by 128 but does not alter the sum of the odd diagonal, which has w = 2.
Given the two-figure and letter code of a tour, this list shows the first two numbers in the top row of its first appearance in the catalogue. For example, the Beverley tour 27a has the first two entries 1,30 and the first occurrence of the Jaenisch tour 00a (denoted 00aA) is under 2,15. It may be noticed that apart from the newly discovered 00o the cyclic tours, type 00, all begin with 2, i.e. they all have an origin a knight move from a corner.
|
|
|
|