Oblique quaternary symmetry, can also be described as ‘90 degree rotatory symmetry’ or ‘windmill symmetry’. This type of symmetry is possible in knight's tours on square boards of side 4n + 2, that is squares 6×6, 10×10, 14×14, 18×18 and so on. For examples of quaternary tours on these boards see the separate pages dealing with these particular boards.
Here we give examples of tours with this type of symmetry on non-square boards, i.e. with non-square outline shape or with holes, or both. The examples cover all sizes of boards from 16 to 196 cells in steps of 4 cells (the boards of 100 or more cells will be in Part 2). Oblique quaternary symmetry requires a board with a multiple of 4 cells, since each quarter of the tour has the same number of moves, m – 1, and of cells, m. The board and the tour can be seen to be composed of four equal parts each of m cells.
If m is odd then the four parts are arranged on the lattice so that the centre of symmetry is at a point where four cells meet, i.e. the corner of a cell. These central cells may or may not form part of the board (for instance they may be at the centre of a 2×2 hole). If m is even however then the four parts are arranged with centre of symmetry at the centre of a cell, which is necessarily not itself part of the board, so in this case there must be a central hole.
Shaped and holey boards can always be thought of as cut from a containing rectangle: in the case of oblique quaternary symmetry a containing square. If the containing square is of side 4n the number of cells omitted for oblique quaternary symmetry to be possible must be 4 + 8k. In frames of side 4n+2 the number of cells omitted must be 8k (possibly zero). In frames of an odd side the cells omitted must number 1 + 8k.
The 8-cells formed by omitting the centre cell of the 3×3 board can of course be toured, but the tour has octonary symmetry, so is not repeated here. A 12-cell board can be toured with direct quaternary symmetry as was shown by Euler, and 12-cell paths with oblique quaternary symmetry exist, but the cells used do not link together to form a connected board. Thus the smallest board that admits a tour with oblique quaternary symmetry is of 16 cells.
16 and 20 cells
24 cells This board can be regarded as 5×5 with centre hole and with corner cells moved one step diagonally. The four corner cells cannot however be added at the middle of each side, to give a board with octonary symmetry, since this fails to maintain the correct balance of colours when the cells are chequered.
28 cells Within the 6×6 there are 8 oblique quaternary tours without holes covering 28 cells.
32 cells Three boards of assorted shapes.
36 cells Oblique quaternary symmetry is possible on the 6×6 square without voids (see the separate pages on 6×6 tours). Here are some alternative 36-cell boards on which oblique quaternary symmetry is possible. There are a number of others that may be worth investigating.
40 cells Two assorted holey boards.
44 cells Three octonary board-shapes can be formed by adding two cells to each side of the 6×6, making 44 cells, and each of these can be toured in quaternary oblique symmetry.
48 cells According to W. H. Cozens (1940) there are 64 tours with 90° rotary symmetry on the 7×7 centreless board. The first five quaternary tours here are formed from a set of closed quarter-tours of the board by ‘simple-linking’ (a further 15 asymmetric tours can be formed from the same set by this process). These are followed by three other examples.
52 cells Assorted examples. The board used by S.Vatriquant (L'Echiquier 1928) can be thought of as a 6×6 with 4 groups of 4 added round the edges. The anonymous example from Zürcher Illustrierte (15 July 1932) was collected by H.J.R.Murray, who gives an 8×8 example with holes.
56 cells Here are two very different tours of a nice approximately circular board (Jelliss 1998).
60 cells There are ten geometrically distinct ways of removing four cells in 90° rotary symmetry from the 8×8 board, but one of these (cell a knight move from corner) does not admit a tour. The nine cases were all solved by G. L. Moore (1920). The first four are shaped-board solutions, the missing cells being round the edges. The other five are the holey cases. Four have four single holes, while in the last the missing cells merge to form a single central hole. These tours are single examples out of many possible.
This subject has been independently rediscovered several times since Moore's work. Here are some alternative solutions. Archibald Sharp gave an example in Linaludo 1925. W. H. Cozens gave four examples in Fairy Chess Review (vol.8 No.3) 1952. It is possible to form tours in which four maximal paths covering 15 cells of the 4×4 quarter-boards are joined together, as in my example.
64 cells A tour with oblique quaternary symmetry is not possible on the normal chessboard, or on any arrangement of 64 squares centred on a point where four cells meet. Solutions are however possible centred on a single cell, with that cell as a hole. The example from Zurcher Illustrierte 27 Oct 1933 was collected by H.J.R.Murray. The other examples are my own. The last example was constructed by starting with four 4×4 squares and diamonds arrays, joining the circuits up by Vandermonde's method (deleting parallel pairs of moves) and finally joining the four 16-move circuits by using a linkage polygon of 8 alternating deleted and inserted moves forming a star round the central hole (see also the 192-cell solution shown towards the end of Part 2).
68 cells The near-circular board here was formed from the 8×8 chess board by removing corner cells and adding two at the centre of each side, as designed by Paul Byway for ‘Troitzky Chess’ (published in Variant Chess 1997). The augmented cross-shaped board is also of 68 cells.
72 and 76 cells One example of each.
80 cells Three examples on the centreless 9×9 board. The example from Zurcher Illustrierte 17 June 1932 was collected by H.J.R.Murray for his 1942 manuscript.
84 and 88 cells Quaternary symmetry is not posssible on the 80-cell Greek cross board, but becomes possible when four squares are added to make a type of celtic cross, as was pointed out by Ernest Bergholt in the magazine Queen 22 January 1916.
92 cells Another of Murray's examples from Zürcher Illustrierte 25 November 1932, and one of my own with five holes in quincunx formation.
96 cells Two ornamented crosses with central hole.
