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

• Published in 2013
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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.

BibTeX entry

@article{ChocolategamesthatsatisfytheinequalityforandGrundynumbers,
title = {Chocolate games that satisfy the inequality $y \leq \left \lfloor \frac{\{}z{\}}{\{}k{\}} \right\rfloor$ for $k=1,2$ and Grundy numbers},
author = {Shunsuke Nakamura and Ryo Hanafusa and Wataru Ogasa and Takeru Kitagawa and Ryohei Miyadera},
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.},
}