Estimating Total Search Space Size for Specific Piece Sets in Chess

Abstract: Automatic chess problem or puzzle composition typically involves generating and testing various different positions, sometimes using particular piece sets. Once a position has been generated, it is then usually tested for positional legality based on the game rules. However, it is useful to be able to estimate what the search space size for particular piece combinations is to begin with. So if a desirable chess problem was successfully generated by examining 'merely' 100,000 or so positions in a theoretical search space of about 100 billion, this would imply the composing approach used was quite viable and perhaps even impressive. In this article, I explain a method of calculating the size of this search space using a combinatorics and permutations approach. While the mathematics itself may already be established, a precise method and justification of applying it with regard to the chessboard and chess pieces has not been documented, to the best of our knowledge. Additionally, the method could serve as a useful starting point for further estimations of search space size which filter out positions for legality and rotation, depending on how the automatic composer is allowed to place pieces on the board (because this affects its total search space size).