The Squares-and-diamonds and Collini circuits on the chessboard have the property that every move is centrally crossed by another knight move. In Chessics 1984-5 I investigated all such pseudotours on the 8×8 board, calling them crosspatch patterns. I gave examples of tours formed from each of these patterns by simple-linking. These results are reproduced here, apart from the Squares-and-diamonds and Collini cases which are dealt with separately.
There are two types of crosspatch pattern on the 8×8 board: those that consist of straights and those that also contain slants. The straight cases correspond to the ways of covering the reduced 4×4 board (where each 2×2 block is represented by a single cell) with wazir-move circuits.
There are 19 geometrically distinct crosspatch patterns on the 8×8 board. Of which 7 employ only straights and the other 12 employ slants.
The Squares-and-diamonds and Collinian pseudotours are the crosspatch patterns with maximum symmetry (octonary). There are two other types that have quaternary (biaxial) symmetry. One consists of two 4×8 boards filled with edge-hugging circuits. The other biaxial crosspatch pattern consists of four circuits of 16 moves within an H outline. Another straight crosspatch consists of four circuits of 16 moves forming a C outline. The other two straight crosspatches fill a half or quarter of the board with squares and diamonds and the rest with edge-hugging circuits.
Seven of the crosspatch patterns that use slants are symmetric; one with central symmetry, one with a diagonal axis, the other five with lateral axis. One of the five with lateral axis is formed of four 16-move circuits, similar to the H-pattern but with the crossbar of the H moved one rank. The 3×4 pseudotour is a component in ten crosspatch patterns, including all the five asymmetric forms.
An alternative way of classifying the patterns is in terms of the number of circuits involved (shown above the diagrams).
The linkage polygons are shown only when they are of 8 moves, since when they are longer it becomes difficult to see the tour pattern.
From the crosspatch consisting of two 4×8 boards filled with edge-hugging circuits, nine symmetric tours can be formed by simple linking. Six are those formed by pairs of edge-hugging 4×8 (open) tours linked across the horizontal median, as were originally studied by Euler (1759). The other three are shown below.
A tour that keeps within the track of the H-shaped crosspatch is impossible, by any method. I found 15 tours (geometrically distinct) from this pattern by simple-linking, but only one (shown) has a symmetric linkage octagon. Similarly the C-shaped crosspatch yields 41 tours by simple linking, but the example given is the only one that keeps within the C outline. In fact it is the only tour, formed by any method, that will fit within the C-shaped track. Example tours from the other two straight crosspatch cases are also shown above.
The centrosymmetric slant crosspatch generates 13 symmetric tours by simple linking, of which eight consist of a pair of 4×8 tours joined. I show one of these eight and two of the other five cases. The next three diagrams show solutions to the three crosspatch patterns that include two 3×4 patches within a 3×8 area. The first is from H. E. Dudeney (1917) who also gave a variant with the pair of 3×4 tours, and their link, reflected left to right.
The last three diagrams above show tours from the remaining three symmetric crosspatch patterns. The final tour is from A. C. Pearson (1907) and derives from the H-pattern with the crossbar of the H moved one rank. Like the H it is formed of four 16-move circuits, but it cannot be simple-linked; the tour shown uses five deletions.
Finally below we show tours derived from the five asymmetric crosspatch patterns.