Shunsuke Nakamura, Ryo Hanafusa, Wataru Ogasa, Takeru Kitagawa and Ryohei Miyadera

Kwansei Gakuin High School

runners@kwansei.ac.jp

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
, where
are the maximum times you can cut them in each direction.

The coordinates
of the chocolates satisfy the inequalities
for
.

For
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
.

For
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
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.

- Introduction
- Chocolate games that satisfy the inequality .
- Chocolates that satisfy the equality for an arbitrary even number
- Grundy Number of chocolate games

- The Chocolates that satisfy the inequality
- The structure of each row
- The structure of each column
- Some definitions for lists of Grundy numbers.
- The structure of Grundy number

- Beautiful Graphs made by Grundy numbers

- Chocolates that satisfy two inequalities
- Bibliography