If you ignore the issue of legality, the number of unique positions up to symmetry is relatively easy to count using Burnside's lemma
. For a combination of pieces without pawns, the group of symmetries of a chess board is D4. This group has 8 elements:
- horizontal and vertical reflections;
- two diagonal reflections;
- rotation by 90 and 270 degrees;
- rotation by 180 degrees.
The number of unique positions up to symmetry with a chosen combination of pieces is the number of orbits of D8 acting on the set of all possible placements of those pieces on a chess board.
According to Burnside's lemma, this number is equal to the average number of placements fixed by an element of D8.
Let's consider KvK. There are 64*63 ways to place these on a chess board. The number of placements fixed by each element:
- identity: 64 * 63
- horizontal / vertical: 0
- diagonals : 8 * 7 (both Ks must be on the diagonal)
- rotation by 90/270: 0
- rotation by 180: 0
So the number of orbits is (64 * 63 + 2 * 8 * 7) / 8 = 518.
For suicide chess these are all legal. For regular chess obviously not.
To count only legal orbits, we should only look at legal placements. Unfortunately this gets nasty:
- identity: 4 * 60 + 24 * 58 + 36 * 55 = 3612
- diagonals: 2 * 6 + 6 * 5 = 42
So the number of orbits is (3612 + 2 * 42) / 8 = 462.
It should be possible to do this by hand for 3 and 4 pieces, but after that it seems to get too messy.