Rectangular Polyomino Set Weak (1,2)-achievement Games

TL;DR

This paper investigates a combinatorial game involving polyominoes where players alternately mark cells, determining winning strategies for certain sets up to size 4 and exploring infinite polyominoes for broader characterization.

Contribution

It determines winning strategies for polyomino sets up to size 4 and introduces the concept of super winners in infinite polyomino achievement games.

Findings

Winning strategies identified for polyomino sets up to size 4

Introduction of super winners in infinite polyomino games

Characterization of all winning teams up to a certain size

Abstract

In a polyomino set (1,2)-achievement game the maker and the breaker alternately mark one and two previously unmarked cells respectively. The maker's goal is to mark a set of cells congruent to one of a given set of polyominoes. The breaker tries to prevent the maker from achieving his goal. The teams of polyominoes for which the maker has a winning strategy is determined up to size 4. In set achievement games, it is natural to study infinitely large polyominoes. This enables the construction of super winners that characterize all winning teams up to a certain size.