Sprague-Grundy values and complexity for LCTR

TL;DR

This paper analyzes the combinatorial game LCTR, showing that optimal strategies often involve mirroring moves, and introduces efficient methods to compute Sprague-Grundy values, significantly improving previous computational bounds.

Contribution

The paper establishes structural properties of LCTR, such as domesticity and returnability, and presents an $O( ext{log}(n))$ algorithm for computing Sprague-Grundy values, improving over the previous $O(n)$ bound.

Findings

Optimal strategies often involve mirroring moves.

LCTR and Downright are domestic and returnable.

Sprague-Grundy values can be computed in $O( ext{log}(n))$ time.

Abstract

Given an integer partition of $n$, we consider the impartial combinatorial game LCTR in which moves consist of removing either the left column or top row of its Young diagram. We show that for both normal and mis\`ere play, the optimal strategy can consist mostly of mirroring the opponent's moves. We also establish that both LCTR and Downright are domestic as well as returnable, and on the other hand neither tame nor forced. For both games, those structural observations allow for computing the Sprague-Grundy value any position in $O(\log(n))$ time, assuming that the time unit allows for reading an integer, or performing a basic arithmetic operation. This improves on the previously known bound of $O(n)$ due to Ili\'c (2019). We also cover some other complexity measures of both games, such as state-space complexity, and number of leaves and nodes in the corresponding game tree.