Geodesic
Chocolate games that satisfy the inequality $y\leq \lfloor \frac{z}{k} \rfloor$ for $k=1,2$ and Grundy numbers
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.
Classification : 20H15 
@article{VM_2013_15_1_a2,
     author = {Shunsuke Nakamura and Ryo Hanafusa and Wataru Ogasa and Takeru Kitagawa and Ryohei Miyadera},
     title = {Chocolate games that satisfy the inequality $y\leq \lfloor \frac{z}{k} \rfloor$ for $k=1,2$ and {Grundy} numbers},
     journal = {Visual Mathematics},
     publisher = {mathdoc},
     volume = {15},
     number = {1},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VM_2013_15_1_a2/}
}
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AU  - Shunsuke Nakamura
AU  - Ryo Hanafusa
AU  - Wataru Ogasa
AU  - Takeru Kitagawa
AU  - Ryohei Miyadera
TI  - Chocolate games that satisfy the inequality $y\leq \lfloor \frac{z}{k} \rfloor$ for $k=1,2$ and Grundy numbers
JO  - Visual Mathematics
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PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/VM_2013_15_1_a2/
LA  - en
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%A Wataru Ogasa
%A Takeru Kitagawa
%A Ryohei Miyadera
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%J Visual Mathematics
%D 2013
%V 15
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%I mathdoc
%U http://geodesic.mathdoc.fr/item/VM_2013_15_1_a2/
%G en
%F VM_2013_15_1_a2
Shunsuke Nakamura; Ryo Hanafusa; Wataru Ogasa; Takeru Kitagawa; Ryohei Miyadera. Chocolate games that satisfy the inequality $y\leq \lfloor \frac{z}{k} \rfloor$ for $k=1,2$ and Grundy numbers. Visual Mathematics, Tome 15 (2013) no. 1. http://geodesic.mathdoc.fr/item/VM_2013_15_1_a2/