Number of distinct checkmate positions

Question

What is the number of distinct checkmate positions in each of the following categories given the following conditions?

1. We are not interested in whether the position can be forced or not or even what sequences of moves can produce the positions. Similarly, these positions are irrespective of player color.

2. Promotion is only allowed if it will produce a set of pieces not possible from the initial set like 3 knights.

3. Redundant pieces like duplicate attackers and other pieces not contributing to the mate are not present.

Then how many exist in each of these categories?

1. A checkmate can exist on a corner, edge or the center, that is to say with 3 adjacent squares, 5, or 8.

2. Let n denote the number of pieces on the board excluding kings. Consider the number of partitions in 2 parts of n. Except for the single case of giving all of the pieces to one player this is the number of ways to divide the pieces between black and white. Of course this doesn't account for what the pieces are, but the question is only about the total number not what the positions actually are.

3. Then what is the number of checkmates for each relevant partition of each n subdivided by corner, edge, or center? Individual solutions for n=1 up to n=30 as much as possible (not that any positions with 32 pieces would satisfy condition #3. It is probably satisfied only for much smaller n). Although it is obvious I should also mention that for n>15 the real number of relevant partitions is smaller because neither player can have >16 pieces.

Again it doesn't matter whether the solution is constructive or not. I suppose a full solution at all is unlikely and a constructive solution would a wonderful bonus.

Answer 1

I haven’t come across this question before. I don’t have the time to execute it, but hope these remarks and suggested approach are helpful.

First let’s clarify: (1) without loss of generality, white is mating black (2) position must be legal - in particularly white king must be on the board even if it doesn’t otherwise contribute to the mate. (3) legitimately promoted non-standard material is fine. (4) position must be minimal in that no subset of pieces can be removed to allow a position which still satisfies 1,2&3 above.

An algorithm could involve putting bK and the checking unit(s). Then consider all the flights, and the different ways they might be blocked. If a black unit might also interfere with check, then that interference needs to be covered. Then can find all the positions for wK if it’s not already placed. This gives a long list of positions which can be checked briefly for legality.

Mostly the positions will be left/right symmetric but there are a very few involving castling.

There are also some quite subtle positions where the position is only mate if we cannot prove that en passant is legal defence. Even some involving dead position rule.