Estimating Total Search Space Size for Specific Piece Sets in Chess

TL;DR

This paper presents a combinatorial method to estimate the size of the search space for specific chess piece sets, aiding in evaluating the feasibility of automatic chess problem generation.

Contribution

It introduces a precise, documented approach for calculating chess position search space sizes considering piece sets and legality constraints.

Findings

Provides a mathematical framework for search space estimation

Clarifies the impact of piece placement rules on search space size

Lays groundwork for optimizing automatic chess problem generation

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).