Sprague-Grundy values and complexity for LCTR

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 misere 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 Ilic (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.(c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).