Chocolate games that are variants of nim and interesting graphs made by these games.

Shunsuke Nakamura,

Daisuke Minematsu,

Takeru Kitagawa,

Youichiro Naito,

Ryohei Fujii,

Takuto Hieda, and

Ryohei Miyadera

Abstract:

We study chocolate games that are variants of a game of Nim. In this article you can cut the chocolate in 3 directions, and we represent the chocolates with coordinates $\{x, y, z\}$ , where $x,y,z$ are the maximum times you can cut it in each direction. The coordinates $\{x, y, z\}$ satisfy the inequality $y\leq \lfloor z/k \rfloor$ for a fixed natural number $k$ . For $k=2$ the authors discovered a formula for loser' s states of the chocolate. For $k$ =1 the authors made predictions for the formulas for loser' s states, although they have not managed to prove them. They also present some interesting graphs made by the set of L states of the chocolate games for $k = 1,2,3,4,5$ .