An upper bound for the number of chess diagrams without promotion

TL;DR

This paper improves the upper bound on the number of legal chess diagrams without promotion from 2×10^40 to less than 4×10^37 by introducing a graph-based method to analyze pawn arrangements.

Contribution

It introduces a novel graph-based approach and a new classification of pawn arrangements to tighten the upper bound on legal chess diagrams.

Findings

Upper bound on diagrams reduced to less than 4×10^37

Graph and class-based methods effectively bound pawn configurations

Improved estimate over previous 2015 result

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.