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

The authors introduce the impartial game of the generalized Ry\=u\=o 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 Ry\=u\=o for a given natural number $p$. The generalized Ry\=u\=o 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 Ry\=u\=o for $p$ is a Ry\=u\=o, i.e., a promoted hisha piece of Japanese chess. A Ry\=u\=o combines the power of the rook and the king in Western chess. The generalized Ry\=u\=o 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 Ry\=u\=o 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 Ry\=u\=o Nim for $p$ with a pass move. The generalized Ry\=u\=o 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 Ry\=u\=o Nim that restricted the diagonal and side movement. Moreover, we extended the Ry\=u\=o Nim dimension to the $n$-dimension.