Cannibal Animal Games: a new variant of Tic-Tac-Toe

TL;DR

This paper introduces the cannibal animal game, a variant of Tic-Tac-Toe played on an infinite grid, and develops new strategies and tools to classify animals as cannibals or non-cannibals, advancing understanding of combinatorial game theory.

Contribution

The paper presents new analytical tools and strategies, such as the bounding strategy and punching lemma, for classifying animals in the cannibal animal game.

Findings

Developed the bounding strategy for animal classification.

Introduced the punching lemma as a new analytical tool.

Showed the effectiveness of the pairing strategy in the game.

Abstract

This paper presents a new partial two-player game, called the \emph{cannibal animal game}, which is a variant of Tic-Tac-Toe. The game is played on the infinite grid, where in each round a player chooses and occupies free cells. The first player Alice can occupy a cell in each turn and wins if she occupies a set of cells, the union of a subset of which is a translated, reflected and/or rotated copy of a previously agreed upon polyomino $P$ (called an \emph{animal}). The objective of the second player Bob is to prevent Alice from creating her animal by occupying in each round a translated, reflected and/or rotated copy of $P$. An animal is a \emph{cannibal} if Bob has a winning strategy, and a \emph{non-cannibal} otherwise. This paper presents some new tools, such as the \emph{bounding strategy} and the \emph{punching lemma}, to classify animals into cannibals or non-cannibals. We also show that the \emph{pairing strategy} works for this problem.