High-Precision Estimation of the State-Space Complexity of Shogi via the Monte Carlo Method

TL;DR

This paper introduces a high-precision Monte Carlo-based method to estimate the total number of reachable positions in Shogi, significantly narrowing previous uncertainty bounds.

Contribution

It presents a novel reachability test using reverse search to accurately estimate Shogi's state-space complexity with billions of samples.

Findings

Estimated Shogi's legal positions as approximately 6.55 x 10^68.

Applied method to Mini Shogi, estimating its complexity as about 2.38 x 10^18.

Reduced search effort by using reverse search to KK positions instead of single-target backward search.

Abstract

Determining the state-space complexity of the game of Shogi (Japanese Chess) has been a challenging problem, with previous combinatorial estimates leaving a gap of five orders of magnitude ($10^{64}$ to $10^{69}$). This large gap arises from the difficulty of distinguishing Shogi positions legally reachable from the initial position among the vast number of valid board configurations. In this paper, we present a high-precision statistical estimation of the number of reachable positions in Shogi. Our method combines Monte Carlo sampling with a novel reachability test that utilizes a reverse search toward a set of "King-King only" (KK) positions, rather than a single-target backward search to the single initial position. This approach significantly reduces the search effort for determining unreachability. Based on a sample of 5 billion positions, we estimated the number of legal positions in Shogi to be $6.55 \times 10^{68}$ (to three significant digits) with a $3\sigma$ confidence level, substantially improving upon previously known bounds. We also applied this method to Mini Shogi, determining its complexity to be approximately $2.38 \times 10^{18}$.