Chocolate games that satisfy the inequality for k=1,2 and Grundy numbers

Shunsuke Nakamura, Ryo Hanafusa, Wataru Ogasa, Takeru Kitagawa and Ryohei Miyadera

runners@kwansei.ac.jp

 

Abstract:

We study chocolate games that are variants of a game of Nim. We can cut the chocolate games in 3 directions, and we represent the chocolates with coordinates $\{x,y,z\}$ , where $x,y,z$ are the maximum times you can cut them in each direction.
The coordinates $\{x,y,z\}$ of the chocolates satisfy the inequalities $y\leq \lfloor \frac{z}{k} \rfloor$ for $k = 1,2$ .
For $k = 2$ we prove a theorem for the L-state (loser's state), and the proof of this theorem can be easily generalized to the case of an arbitrary even number $k$ .
For $k = 1$ we prove a theorem for the L-state (loser's state), and we need the theory of Grundy numbers to prove the theorem. The generalization of the case of $k = 1$ to the case of an arbitrary odd number is an open problem. The authors present beautiful graphs made by Grundy numbers of these chocolate games.