Achievement Games on a One-dimensional Board

Abstract

Achievement games are usually played and studied for polynominos which are commonly called animals, and it is known whether each animal is a winner or a loser, except for one animal called Snaky. In this paper, we study achievement games for unconnected shapes which we call creatures. We mostly discuss them on a one-dimensional board, because they exhibit interesting behavior even in this simple setting. We determined whether a creature is a winner or a loser for those composed of no more than three stones. Furthermore, we proved there is an arbitrary large paving winner.