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arXiv · 2604.06189

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

Sotaro Ishii·Tetsuro Tanaka
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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 $2.38 \times 10^{18}$.

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Sotaro Ishii, Tetsuro Tanaka. 2026-02-24. High-Precision Estimation of the State-Space Complexity of Shogi via the Monte Carlo Method. https://arxiv.org/abs/2604.06189

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