For a description of birotary symmetry, see the Introduction to the page on Shaped boards. Here we give examples of tours with this type of symmetry on boards with one or more holes. The examples cover most boards from 16 to 196 cells in steps of 4 cells. On boards with holes the number of moves, m, in a quarter of the tour can be odd or even.
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 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. If the outline of the board is an odd-sided square then the number of cells omitted must be 1 + 8k.
The 8-cell board 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 biaxial symmetry, with diagonal axes, as was shown by Euler, and 12-cell paths with birotary symmetry exist, but the cells used do not link together to form a connected board. Thus the smallest board that admits a tour with birotary symmetry is of 16 cells.
I have only found the one tour with birotary symmetry on a holey board of 20 cells, as shown here (Jelliss 2013). This has four single-cell holes in skew formation. The similar board with a 2×2 central hole only admits a quaternary tour with biaxial symmetry (with diagonal axes).
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.
Two tours on a board with four single holes in a skew pattern (Jelliss 2013). As for the 20 cell case the similar board with a 2×2 central hole has a tour with biaxial symmetry.
Three boards of assorted shapes.
Birotary symmetry is possible on the 6×6 square without voids (see the separate pages on 6×6 tours). Here are some alternative 36-cell holey boards on which birotary symmetry is possible. Harikrishna's #45 (1871) uses the same board as the first four tours here, but his tour is asymmetric.
Here are two boards with four single-cell holes.
There are a number of other 36-cell boards that may be worth investigating.
Two assorted holey boards.
Here is a quaternary board with four quaternary tours (Jelliss 2013).
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.
The example with four holes, from Zürcher Illustrierte (15 July 1932), was collected by H. J. R. Murray, who gives an 8×8 example of his own with five holes (Murray 1942).
Here are two very different tours of a nice approximately circular board (Jelliss 1998).
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 four with missing cells round the edges are shown in the page on Shaped boards. The four with four single-cell holes are shown here. The two extra examples are by W. H. Cozens Fairy Chess Review (vol.8 No.3) 1952. All the tours in this section are single examples out of many possible.
In the final case the four missing cells merge to form a single central hole. These examples are by Moore 1920, Cozens 1952 and Jelliss 1985.
A tour with birotary symmetry is not possible on the normal chessboard, nor 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).
A tour (Jelliss 2013) on a board formed of four 4×4 boards and four single cells arramged round a 2×2 space.
Kraitchik 1927.
A tour on a 10×10 board with a 24-cell central clearance (Jelliss 2013).
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.
I add an example of my own (Jelliss 2013) with a size 2 Greek Cross in the centre,
where the hole is shown by a dot in this case to avoid obscuring the centre pattern.
This and the 100 cell example illustrate how tours on larger even areas can often be easily constructed by joining together four tours on small boards arranged round a central hole.
Jelliss 1998
One of my own (Jelliss 1996) with five holes in quincunx formation.
Two ornamented crosses with central hole.
Quaternary symmetry is possible on the square board 10×10 (see the separate pages on 10×10 tours for numerous examples). Here is a simple holey example (Jelliss 2013).
Jelliss 2003
The first tour is from Denken und Raten 26 July 1931, recorded by Murray, formed by omitting 36 cells from the 12×12. The second 108 example was constructed by accident when aiming for 112. This and the two others are my own work (Jelliss 2003).
Jelliss 2003
Two more tours collected by Murray from Denken und Raten both 25 August 1929, omitting 28 cells from the 12×12 in various places.
The 120 is by Kraitchik 1927.
A tour using a board made up from four 4×8 boards with overlapping corners, leaving a 6×6 space (Jelliss 2013).
It may be worth noting here that in constructing tours like this 128-cell example the central cross-shaped hole has to be plus-shaped rather than X-shaped to ensure that the balance between black and white cells (when the board is chequered) is maintained, as is required in all closed knight tours. Trying to construct a tour on a board where the colours are not in balance is a exercise in futility.
Clockface design on a 12×12 Board with 12 vacant cells in a circle, forming eight holes (Jelliss 2013).
The centres of the 12 void cells are all exactly the same distance from the centre point,
namely (√50)/2 = 5/√2 = 3.5355 if the side of a cell is taken as unit.
The example for 136 was constructed by joining four versions of a 3×7 tour with circuits on the central 7×7 area.
The 140 example is from Kraitchik 1927.
Quaternary symmetry is impossible on the 12×12 square of course. The first example here simply stitches together four copies of a 6×6 tour around a central single-cell hole. The second example uses a board with octonary symmetry.
‘Superchess’ was a variety of chess on this enlarged board that was described in Variant Chess 1994 (Jelliss 1994).
NOTE: Apart from the last 196 board, no further examples on boards of size 8n + 4 (i.e. 4 times an odd number) are given here since they are easily constructed, and the same frame admits similar tours without holes.
Jelliss 2003
This 160-cell tour (on a board of size 13² – 3²) was constructed by first joining four 5×5 corner-to-corner tours together (their corners overlapping), then joining in four 4×4 sets of squares and diamonds in the four corners by Vandermonde's method (Jelliss, undated).
The 168-cell example by Kraitchik (1927) uses the 13×13 with missing centre cell.
The 176-cell skewed-cross example (Jelliss 2003) makes visual use of the arithmetic decomposition 176 = 4×4×11 = 4×(4×8 + 4×3), being a composite of four tours 4×8 and four tours 4×3 arranged around a central single-cell hole.
The 184 example (Jelliss 2003) uses the decomposition 184 = 4×46 = 4×(16 + 30) = 4×(4×4 + 5×6). Thus, similar to the 160 case, four 5×6 open tours were first joined together, then the squares and diamonds in the four corner 4×4s were joined in by Vandermonde's method.
The central pattern of the 192-cell example is the 64-cell example shown above extended to fill 8 more 4×4 areas by the braid method. Tours with quaternary symmetry are possible on the 196-cell square 14×14 (see the separate pages on square tours for examples). The example shown here, in similar fashion to the first 144-cell example uses the fact that 196 = 4×(7²).
