Combinatorial game associated to the one dimensional Schelling’s model of social segregation

In this paper we consider a finite one-dimensional lattice with N= 2 n+ 1 sites such that one of them is empty and the others have a black or white token. There are two players (one for each color), such that step by step alternately they move one of their tokens to the empty site trying to obtain a connected configuration. This game is related with the Schelling’s social segregation model, where colors represent two different populations such that each one tries to take up a position with more neighbors as itself (same color). In this work we study strategies to play the game as well as their relation with the associated Schelling’s one-dimensional case (line and cycle graphs).