Chocolate games that satisfy the inequality $y\leq \lfloor \frac{z}{k} \rfloor$ for $k=1,2$ and Grundy numbers

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

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.