NON-LOSING STRATEGIES FOR CASTELLAN

Abstract

Non-losing strategies in games ensure a player can play in a manner in which, though they may not win, they do not lose. This thesis explores non-losing strategies for a restricted version of Castellan. Castellan is a game where each player attempts to best the other by using walls connected to towers to enclose regions with the most towers bordering them. The question was narrowed to games with a rectangular layout. Research was conducted using a program written to enumerate games of relatively small size, ones having fewer than five unit regions. After observing outcomes of the computer program, conjectures were formed and lemmas proved. In the end, it was found that with the restricted rules, the player to place the first piece has a non-losing strategy for games where the rows and columns both have an odd number of tower locations, or where exclusively the rows or towers have an odd number of tower locations. Lemmas and corollaries that are proved are used to support this fact.