2022/07/12 by Eric Gottlieb, Gottlieb, Eric, Matjaž Krnc +3 · 2 citations
Computer Science · #05A17 #91A46 #91A68 #Advanced Algebra and Logic #Combinatorics (math.CO) #Computational Complexity (cs.CC) #F.2.2 #FOS: Computer and information sciences #FOS: Mathematics #G.2.1
paper · pdf · doi:10.48550/arxiv.2207.05599
openalex publication_date 2022/07/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 misère 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 Ilić (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.