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The authors introduce the impartial game of the generalized Ry\=u\=o Nim, avariant of the classical game of Wythoff Nim. In the latter game, two playerstake turns in moving a single queen on a large chessboard, attempting to be thefirst to put her in the upper left corner, position $(0,0)$. Instead of thequeen used in Wythoff Nim, we use the generalized Ry\=u\=o for a given naturalnumber $p$. The generalized Ry\=u\=o for $p$ can be moved horizontally andvertically, 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 $1 \leq s \leqx, 1 \leq t \leq y \textit{and} s+t \leq p-1$. When $p$ is $3$, the generalizedRy\=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. Thegeneralized Ry\=u\=o Nim for $p$ is mathematically the same as the Nim with twopiles of counters in which a player may take any number from either heap, and aplayer 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 naturalnumber. 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 movehas simple formulas for Grundy numbers. This is not the case after theintroduction of a pass move, but it still has simple formulas for the previousplayer's positions. We also study the Ry\=u\=o Nim that restricted the diagonaland side movement. Moreover, we extended the Ry\=u\=o Nim dimension to the$n$-dimension.

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