On the number of positions in chess without promotion

Abstract

The number of different legal positions in chess is usually estimated to be between $10^{40}$ and $10^{50}$. Within this range, the best upper bound $10^{46}$ is some orders of magnitude bigger than the estimate $5 \times 10^{42}$ made by Claude Shannon in his seminal 1950 paper, which is usually considered by computer scientists to be a better approximation. We improve Shannon’s estimate and show that the number of positions where any number of chessmen may have been captured but no promotion has occured is bounded from above by $2 \times 10^{40}$. The actual number should be quite a bit smaller than that and outline possible ways towards improving our result.