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

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.

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