arXiv subjects

Ryohei Miyadera

Publications and source records attributed to Ryohei Miyadera.

21 records · Page 2Linked to original sources

arXiv2017

Impartial Triangular Chocolate Bar Games

Chocolate bar games are variants of the game of Nim in which the goal is to leave your opponent with the single bitter part of the chocolate bar. The rectangular chocolate bar game is a thinly disguised form of classical multi-heap Nim. In this work, we investigate the mathematical structure of triangular chocolate bar games in which the triangular chocolate bar can be cut in three directions. In the triangular chocolate bar game, a position is a $\mathcal{P}$-position if and only if $x \oplus y \oplus z = 0$, where the numbers $x,y,z$ stand for the maximum number of times that the chocolate bar can be cut in each direction. Moreover, the Grundy number of a position $(x,y,z)$ is not always equal to $x \oplus y \oplus z $, and a generic formula for Grundy numbers in not known. Therefore, the mathematical structure of triangular chocolate bar game is different from that of classical Nim.

math.CO↗

arXiv2017

Grundy Numbers of Impartial Chocolate Bar Games

Chocolate bar games are variants of the CHOMP game in which the goal is to leave your opponent with the single bitter part of the chocolate. In this paper, we investigate step chocolate bars whose widths are determined by a fixed function of the horizontal distance from the bitter square. When the width of chocolate bar is proportional to the distance from the bitter square and the constant of proportionality is even, the authors have already proved that the Grundy number of this chocolate bar is $(m-1) \oplus (n-1)$, where $m$ is is the largest width of the chocolate and $n$ is the longest horizontal distance from the bitter part. This result was published in a mathematics journal(Integers,15, 2015). On the other hand, if the constant of proportionality is odd, the Grundy number of this chocolate bar is not $(m-1) \oplus (n-1)$. Therefore, it is natural to look for a necessary and sufficient condition for chocolate bars to have the Grundy number that is equal to $(m-1) \oplus (n-1)$, where $m$ is the largest width of the chocolate and $n$ is the longest horizontal distance from the bitter part. In the first part of the present paper, the authors present this necessary and sufficient condition. Next, we modified the condition that the Grundy number that is equal to $(m-1) \oplus (n-1)$, and we studied a necessary and sufficient condition for chocolate bars to have Grundy number that is equal to $((m-1) \oplus (n-1+s))-s$, where $m$ is is the largest width of the chocolate and $n$ is the longest horizontal distance from the bitter part. We present this necessary and sufficient condition in the second part of this paper.

math.CO↗

arXiv2017

Ryuo Nim: A Variant of the classical game of Wythoff Nim

The authors introduce the impartial game of the generalized Ryūō Nim, a variant of the classical game of Wythoff Nim. In the latter game, two players take turns in moving a single queen on a large chessboard, attempting to be the first to put her in the upper left corner, position $(0,0)$. Instead of the queen used in Wythoff Nim, we use the generalized Ryūō for a given natural number $p$. The generalized Ryūō for $p$ can be moved horizontally and vertically, as far as one wants. It also can be moved diagonally from $(x,y)$ to $(x-s,y-t)$, where $s,t$ are non-negative integers such that $1 \leq s \leq x, 1 \leq t \leq y \textit{and} s+t \leq p-1$. When $p$ is $3$, the generalized Ryūō for $p$ is a Ryūō, i.e., a promoted hisha piece of Japanese chess. A Ryūō combines the power of the rook and the king in Western chess. The generalized Ryūō Nim for $p$ is mathematically the same as the Nim with two piles of counters in which a player may take any number from either heap, and a player may also simultaneously remove $s$ counters from either of the piles and $t$ counters from the other, where $s+t \leq p-1$ and $p$ is a given natural number. The Grundy number of the generalized Ryūō Nim for $p$ is given by $\bmod(x+y,p) + p(\lfloor \frac{x}{p} \rfloor \oplus \lfloor \frac{y}{p}\rfloor)$. The authors also study the generalized Ryūō Nim for $p$ with a pass move. The generalized Ryūō Nim for $p$ without a pass move has simple formulas for Grundy numbers. This is not the case after the introduction of a pass move, but it still has simple formulas for the previous player's positions. We also study the Ryūō Nim that restricted the diagonal and side movement. Moreover, we extended the Ryūō Nim dimension to the $n$-dimension.

math.CO↗