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

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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• http://www.mi.sanu.ac.rs/vismath/miyadera2013/index.html

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BibTeX entry

@article{ChocolategamesthatsatisfytheinequalityforandGrundynumbers,
	key = {ChocolategamesthatsatisfytheinequalityforandGrundynumbers},
	type = {article},
	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.},
	comment = {},
	date_added = {2017-10-18},
	date_published = {2013-04-10},
	urls = {http://www.mi.sanu.ac.rs/vismath/miyadera2013/index.html},
	collections = {Attention-grabbing titles,Food,Games to play with friends},
	url = {http://www.mi.sanu.ac.rs/vismath/miyadera2013/index.html},
	urldate = {2017-10-18},
	year = 2013
}