An upper bound for the number of chess diagrams without promotion
Daniel Gourion
TL;DR
This work bounds the number of legal chess diagrams without promotion by introducing a graph-based, quadruplet representation $\mathcal{P}=(P_w,P_b,p_w,p_b)$ with root $\mathcal{P}_0=(8,8,8,8)$ and edges that correspond to captures. It then defines pawn-arrangement classes—8 files with ordered pawn stacks—so that the combinatorics of pawn placements can be bounded via $\prod_{i=1}^{8}\binom{6}{k_i}$ per class, allowing recursive propagation of bounds from predecessors. After bounding pawns, piece placements are counted with parity constraints for bishops and standard combinatorics, yielding a total bound $n_\mathcal{P}\times m_\mathcal{P}$; a specialized refinement handles diagrams with $2$–$23$ pieces. The result is a substantial tightening of prior estimates, giving an overall upper bound of $<4\times 10^{37}$ diagrams without promotion, with the best-case bound for $24$–$32$ pieces expressed as $3.7422\ldots\times 10^{37}$. The authors also provide verifiability via publicly available code and data and propose extensions to include promotions and additional game features.
Abstract
In 2015, Steinerberger showed that the number of legal chess diagrams without promotion is bounded from above by $2\times 10^{40}$. This number was obtained by restricting both bishops and pawns position and by a precise bound when no chessman has been captured. We improve this estimate and show that the number of legal diagrams is less than $4\times 10^{37}$. To achieve this, we define a graph on the set of diagrams and a notion of class of pawn arrangements, leading to a method for bounding pawn positions with any number of men on the board.