A Variant of Wythoff's Game Defined by Hofstadter's G-Sequence
Kahori Komak, Ryohei Miyadera, Aoi Murakami
TL;DR
This work examines a variant of Wythoff's game with terminal positions $U=\{(x,y): x+y ≤ 2\}$ and a queen-move rule, characterizing its $\mathcal{P}$-positions via floor-based constructions tied to Hofstadter's G-sequence. It provides an explicit description of the $\mathcal{P}$-positions using two families parameterized by $n$ and a recursing function $g(n)$, and reveals two key large-coordinate properties: the Grundy number equals $1$ precisely when a position is a $P$-position of Wythoff's game, and the misere version shares the same large-coordinate structure as Wythoft's. The relation between $g$ and Hofstadter's G-sequence is made precise through $h$, enabling a compact reformulation $g(n)=1-g(h(n-1))$ when $h(n-2)<h(n-1)$. The paper also extends the analysis to misere play and to the sum with a one-pile Nim, showing consistent alignment with Wythoff's $P$-positions in the large-coordinate regime and connecting Beatty-type sequences, Hofstadter dynamics, and Sprague-Grundy theory.
Abstract
In this paper, we study a variant of the classical Wythoff's game. The classical form is played with two piles of stones, from which two players take turns to remove stones from one or both piles. When removing stones from both piles, an equal number must be removed from each. The player who removes the last stone or stones is the winner. Equivalently, we consider a single chess queen placed somewhere on a large grid of squares. Each player can move the queen toward the upper-left corner of the grid, either vertically, horizontally, or diagonally, in any number of steps. The winner is the player who moves the queen into the upper-left corner, the position (0,0) in our coordinate system. We call (0,0) the terminal position of Wythoff's game. In our variant of Wythoff's game, we have a set of positions {(0,0),(1,0),(0,1),(1,1),(2,0),(0,2)} as the terminal set. If a player moves the queen into this terminal set, that player is the winner of the game. The P-positions of this variant are described by the P-positions of Wythoff's game and Hofstadter's G-Sequence. This variant has two remarkable properties. For a position (x,y) with x >= 8 or y >= 8, the Grundy number of the position (x,y) is 1 in this variant if and only if (x,y) is a P-position of Wythoff's game. There is another remarkable property.For a position (x,y) with x >= 8 or y >= 8, (x,y) is a P-position of of the misere version of this variant if and only if (x,y) is a P-position of of Wythoff's game.