Strategies for playing closing octagons game in some situations

Abstract

Closing Octagons game is a combinatorial game for two players that we modify from Dots and Boxes game and Dots and Hexagons game. The game starts with an m × n array of octagons such that every two adjacent octagons has one common side and 0 points for both players. Two players alternately turn by the following rules. (i) A player moves by coloring one side of an octagon. (ii) A player who colors the eighth side of k octagons earns k points and takes one more move. The game ends when every side of octagons has been colored and a player having the most points wins. This game is formulated into the new game using graphs. In order to analyze the game, more rules are added and the game with these additional rules is called normal game. In this thesis, we analyze patterns of the game and give strategies on how to play the game in several situations.