100 and 104 cells Quaternary symmetry is possible on the square board 10×10 (see the separate pages on 10×10 tours for numerous examples). The example here is just to show that there are other interesting 100-cell shapes besides squares.
108 cells Two tours from Denken und Raten 15 March 1931 and 26 July 1931, recorded by Murray, formed by omitting 36 cells from the 12×12 in various places.
108 and 112 cells This extra 108 example was constructed by accident when aiming for 112. These are my own work (Jelliss 2003).
116 cells Two more tours collected by Murray from Denken und Raten both 25 August 1929, omitting 28 cells from the 12×12 in various places.
120 and 124 cells The 120 is by Kraitchik 1927. The 124 is another example from Denken und Raten 14 June 1931 (can be regarded as 10×10 + 24 or 12×12 – 20).
128 and 132 cells It may be worth noting here that in constructing tours like the 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.
136 and 140 cells 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.
144 cells 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.
148 cells ‘Superchess’ was a variety of chess on this enlarged board that was described in Variant Chess 1994.
152 and 156 cells The central pattern in my 156 quaternary example is adapted from two designs used by Edward Falkener (1892) in his 160-cell tours (his tours numbers 9 and 15).
160 and 164 cells The 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. The 164 example is after Falkener's (1892) near-symmetric design number 16 on a 160-cell ‘Four-Handed Chess’ board (on which quaternary symmetry is impossible). The four extra cells in the concave corners of the cross make quaternary symmetry possible.
168 and 172 cells The 168-cell example by Kraitchik (1927) uses the 13×13 with missing centre cell. The 172 cell example is 14×14 minus 24 (six at each corner).
176 and 180 cells The 176-cell skewed-cross example 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 180-cell tour uses 180 = 14² – 4×(2²).
184 and 188 cells The 184 example 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.
192 and 196 cells The central pattern of the 192-cell example is the 64-cell example shown in Part 1 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).
I hope that, rather than simply reproduce the examples shown in this collection readers will be encouraged to construct their own tours, using pen or pencil on squared paper. Many more artistic and quirky patterns can be found. It can be an enjoyable recreation. Send me your best examples (preferably as monochromatic gif images). The diagrams shown here have all been drawn using the drawing facility in Lotus WordPro (with the grid setting showing dots at 0.1 inch separation). It is just a matter of joining up the dots! In the case of tours with quaternary symmetry it is only necessary to draw a quarter of the tour on the board to begin with. The rest of the diagram can then be completed by copying and rotating the first quarter.