Related papers: Compound Node-Kayles on Paths

Node-Kayles is a well-known impartial combinatorial game played on graphs, where players alternately select a vertex and remove it along with its neighbors. By the Sprague-Grundy theorem, every position of an impartial game corresponds to a…

Combinatorial Game Theory typically studies sequential rulesets with perfect information where two players alternate moves. There are rulesets with {\em entailing moves} that break the alternating play axiom and/or restrict the other…

This is an introduction into John Conway's beautiful Combinatorial Game Theory, providing precise statements and detailed proofs for the fundamental parts of his theory. (1) Combinatorial game theory, (2) the GROUP of games, (3) the FIELD…

We analyze the computational complexity of several new variants of edge-matching puzzles. First we analyze inequality (instead of equality) constraints between adjacent tiles, proving the problem NP-complete for strict inequalities but…

Number games play a central role in alternating normal play combinatorial game theory due to their real-number-like properties (Conway 1976). Here we undertake a critical re-examination: we begin with integer and dyadic games and identify…

Combinatorial Game Theory has also been called `additive game theory', whenever the analysis involves sums of independent game components. Such {\em disjunctive sums} invoke comparison between games, which allows abstract values to be…

Using coalgebraic methods, we extend Conway's theory of games to possibly non-terminating, i.e. non-wellfounded games (hypergames). We take the view that a play which goes on forever is a draw, and hence rather than focussing on winning…

Impartial subtraction games on the nonnegative integers have been studied by many and discussed in detail in for example the remarkable work Winning Ways by Conway, Berlekamp and Guy. We describe how comply variations of these games,…

A class of discrete Bidding Combinatorial Games that generalize alternating normal play was introduced by Kant, Larsson, Rai, and Upasany (2022). The major questions concerning optimal outcomes were resolved. By generalizing standard game…

Arc-Kayles is a game where two players alternate removing two adjacent vertices until no move is left, the winner being the player who played the last move. Introduced in 1978, its computational complexity is still open. More recently,…

We present a computational formulation for the approximate version of several variational inequality problems, investigating their computational complexity and establishing PPAD-completeness. Examining applications in computational game…

This paper investigates the popular card game UNO from the viewpoint of algorithmic combinatorial game theory. We define simple and concise mathematical models for the game, including both cooperative and uncooperative versions, and analyze…

We define and give results on the game NecklaceNim NN($n$,$k$) which is PathNim PN($n$,$k$) with an additional move allowed on the end vertices. This game arises as a sub-game in the context of solving CircularNim CN($n$,$k$) when $k-2$…

We analyze Solo Chess puzzles, where the input is an $n \times n$ board containing some standard Chess pieces of the same color, and the goal is to make a sequence of capture moves to reduce down to a single piece. Prior work analyzes this…

The New York Times (NYT) games have found widespread popularity in recent years and reportedly account for an increasing fraction of the newspaper's readership. In this paper, we bring the computational lens to the study of New York Times…

We announce misere-play solutions to several previously-unsolved combinatorial games. The solutions are described in terms of misere quotients--commutative monoids that encode the additive structure of specific misere-play games. We also…

The classic result by Fortune, Hopcroft, and Wyllie [TCS~'80] states that the directed disjoint paths problem is NP-complete even for two pairs of terminals. Extending this well-known result, we show that the directed disjoint paths problem…

We introduce a new family of one-player games, involving the movement of coins from one configuration to another. Moves are restricted so that a coin can be placed only in a position that is adjacent to at least two other coins. The goal of…

What does it mean today to study a problem from a computational point of view? We focus on parameterized complexity and on Column 16 "Graph Restrictions and Their Effect" of D. S. Johnson's Ongoing guide, where several puzzles were proposed…

In an investigation of the applications of Combinatorial Game Theory to chess, we construct novel mutual Zugzwang positions, explain an otherwise mysterious pawn endgame from "A Guide to Chess Endings" (Euwe and Hooper), show positions…