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

• Published in: arXiv.org 2017-11
• Main Authors: Miyadera, Ryohei, Tokuni, Yuki, Nakaya, Yushi, Fukui, Masanori, Abuku, Tomoaki, Suetsugu, Koki
• Format: Article
• Language: English
• Subjects: Chess; Games; Integers
• ISSN: 2331-8422

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.