# Construction, observation and knowledge abstraction for go endgames on small boards

## TL;DR

This paper explores optimal moves in small Go boards to understand game properties and difficulty factors.

## Contribution

The paper introduces methods to construct Go endgame databases and identifies patterns in optimal moves based on board size and rules.

## Key findings

- Optimal game values differ for even and odd board sizes, regardless of cycle rules.
- Allowing cycles equalizes player strength on even boards but not on odd ones.
- A formula was developed to predict positions with specific difficulty levels.

## Abstract

A Go endgame database consists of optimal game values and moves for every legal arrangement of no more than S pieces on an N by N board. This paper describes methods for constructing such databases when \documentclass[12pt]{minimal}
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\begin{document}$$1 < N \le 5$$\end{document}1<N≤5 and \documentclass[12pt]{minimal}
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\begin{document}$$S = N^2$$\end{document}S=N2. When cycles of plies with lengths greater than 4 are encountered, two rules, one allowing cycles and the other disallowing them, are implemented. Observations and knowledge are obtained for these endgames, which may elucidate the fundamental properties of the popular game Go. First, the optimal game values are different when N is even and odd, regardless of whether the repetition of positions is allowed. When N is odd, the first player can occupy the whole board, while this is not the case when N is even. Second, allowing cycles makes the first and second players equal in strength when N is even, whereas the first player always dominates when N is odd. Using the state-of-the-art open-source deep learning Go engine KataGo to correctly solve a given position as an indicator, factors affecting level of difficulty are found, including the distributions of the optimal game values among all legal plies and the cardinality and values of the true optimal plies. A simple formula is designed that works on more than 10% of the positions so that positions with a given level of difficulty can be found with a high probability.