# 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
- Added on
2017-10-18

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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.

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### 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}, url = {http://www.mi.sanu.ac.rs/vismath/miyadera2013/index.html}, urldate = {2017-10-18}, year = 2013, 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.}, comment = {}, collections = {attention-grabbing-titles,food,games-to-play-with-friends} }