{"title":"Chess is hard even for a single player","authors":"","doi":"10.1016/j.tcs.2024.114726","DOIUrl":null,"url":null,"abstract":"<div><p>We introduce a generalization of “Solo Chess”, a single-player variant of the game that can be played on chess.com. The standard version of the game is played on a regular 8 × 8 chessboard by a single player, with only white pieces, using the following rules: every move must capture a piece, no piece may capture more than 2 times, and if there is a King on the board, it must be the final piece. The goal is to clear the board, i.e., make a sequence of captures after which only one piece is left.</p><p>We generalize this game to unbounded boards with <em>n</em> pieces, each of which have a given number of captures that they are permitted to make. We show that <span>Generalized Solo Chess</span> is <span>NP</span>-complete, even when it is played by only rooks that have at most two captures remaining. It also turns out to be <span>NP</span>-complete even when every piece is a queen with exactly two captures remaining in the initial configuration. In contrast, we show that solvable instances of <span>Generalized Solo Chess</span> can be completely characterized when the game is: a) played by rooks on a one-dimensional board, and b) played by pawns with two captures left on a 2D board.</p><p>Inspired by <span>Generalized Solo Chess</span>, we also introduce the <span>Graph Capture Game</span>, which involves clearing a graph of tokens via captures along edges. This game subsumes <span>Generalized Solo Chess</span> played by knights. We show that the <span>Graph Capture Game</span> is <span>NP</span>-complete for undirected graphs and DAGs.</p></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Theoretical Computer Science","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0304397524003438","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
Abstract
We introduce a generalization of “Solo Chess”, a single-player variant of the game that can be played on chess.com. The standard version of the game is played on a regular 8 × 8 chessboard by a single player, with only white pieces, using the following rules: every move must capture a piece, no piece may capture more than 2 times, and if there is a King on the board, it must be the final piece. The goal is to clear the board, i.e., make a sequence of captures after which only one piece is left.
We generalize this game to unbounded boards with n pieces, each of which have a given number of captures that they are permitted to make. We show that Generalized Solo Chess is NP-complete, even when it is played by only rooks that have at most two captures remaining. It also turns out to be NP-complete even when every piece is a queen with exactly two captures remaining in the initial configuration. In contrast, we show that solvable instances of Generalized Solo Chess can be completely characterized when the game is: a) played by rooks on a one-dimensional board, and b) played by pawns with two captures left on a 2D board.
Inspired by Generalized Solo Chess, we also introduce the Graph Capture Game, which involves clearing a graph of tokens via captures along edges. This game subsumes Generalized Solo Chess played by knights. We show that the Graph Capture Game is NP-complete for undirected graphs and DAGs.
期刊介绍:
Theoretical Computer Science is mathematical and abstract in spirit, but it derives its motivation from practical and everyday computation. Its aim is to understand the nature of computation and, as a consequence of this understanding, provide more efficient methodologies. All papers introducing or studying mathematical, logic and formal concepts and methods are welcome, provided that their motivation is clearly drawn from the field of computing.