Complementary five-leaper tours with rotational symmetry (John)
The "five-leaper", a piece which jumps a distance of exactly five squares in any direction, has exactly four moves available to it from any square on a normal 8x8 chessboard, and a pair of five-leaper tours are said to be "complementary" if from each square one of them uses the two moves that the other does not. In Variant Chess 64, I stated that there were 224 geometrically distinct self-complementary tours with rotational symmetry (a rotation of 90 degrees gave a complementary tour, a further rotation of 90 degrees gave the original tour again). The file fiveleapertours.txt (approximately 390 Kb) presents the computer results on which this assertion was made.
In this file, each of the 224 geometrically distinct tours appears eight times (it can be flipped about the leading diagonal, it can be numbered in either direction, it can be rotated by 90 degrees to give the complementary tour, and each of these transformations can be applied independently). Thus tour 1g is tour 1a flipped about the leading diagonal, tours 1c and 1f are tours 1a and 1g numbered in the other direction, and tours 1e, 1d, 1h, and 1b are tours 1a, 1g, 1c, and 1f respectively rotated through 90 degrees and renumbered to put 1 at the top left corner.
In Variant Chess 64, I wrote that these results should be regarded as provisional until they had been confirmed by an independent worker, since if a programming error had caused me to miss a batch there would have been nothing to tell me. This remains true, but what would seem to have been a straightforward and systematic exploration, using a program written for simplicity rather than speed, turned up each of these 224 tours in each of its eight possible presentations and nothing else, and I hope this gives at least some reason to believe that the enumeration was complete and correct.