Generalized Algorithm for Wythoff's Game with Basis Vector $(2^b,2^b)$

Abstract

Wythoff's Game is a variation of Nim in which players may take an equal number of stones from each pile or make valid Nim moves. W. A. Wythoff proved that the set of P-Positions (losing position), C, for Wythoff's Game is given by C := ( kφ , kφ 2 ), ( kφ 2 , kφ ) : k ∈ Z ≥0 [Wyt07]. An open Wythoff problem remains where players make the valid Nim moves or remove kb stones from each pile, where b is a fixed integer. We denote this as the (b, b) game. For example, regular Wythoff's Game is just the (1, 1) game. In 2009, Duchêne and Gravier [DG09] proved an algorithm to generate the set of P-Positions for the (2, 2) game by exploiting the periodic nature of the differences of stones between the two piles modulo 4. We observe similar cyclic behaviour (see definition 3.2) for any b, where b is a power of 2, modulo b 2 , and construct an algorithm to generate the set of P-Positions for this game. Let a be a power of 2. We prove our algorithm works by first showing that it holds for the first a 2 terms in the (a, a) game. Next, we construct an ordered multiset for the (2a, 2a) game from the a 2 terms, and an inductive proof follows. Moreover, we conjecture that all cyclic games require a to be a power of 2, suggesting that there is no similar structure in the generalised (b, b) game where b isn't a power of 2. Future directions for generalising this result would likely utilise numeration systems, particularly the PV numbers.

Self-similarity of $\mathcal{P}$-positions of $(2n+1)$-dimensional Wythoff's game

“…A position is called a P-position if no matter what move the first player takes, the second player always has a winning strategy. In the classic Wythoff's game, for instance, we can easily find that (2,2) and (1,2) are N -position and P-position, respectively. In general, a game position in the classic Wythoff's game is either an N -position or a P-position; see [3].…”

“…Duchêne et al [5] introduced the restriction that one cannot remove more than R tokens from a single heap; meanwhile, Liu, Li and Li [17] allowed removing the same (arbitrarily large) number of tokens from both heaps. In [5], Duchêne et al also proposed the (a, a) game where to remove k tokens from both heaps, k must be a positive multiple of a only integer multiples of a. Aggarwal et al [1] studied the algorithm for (2 b , 2 b ) game. Duchêne et al [7] investigated extensions and restrictions of Wythoff's game having exactly the same set of P-positions as the original game.…”

“…, 1)}. Aggarwal, Geller, Sadhuka and Yu [1] conjectured that the set of P-positions of 3-dimensional Wythoff's game is related to the classic fractal-Sierpinski sponge, i.e. the compact set K ⊂ R 3 satisfying…”

“…Conjecture 2 (Siperpinski sponge conjecture [1]). The set of P-position of 3-dimensional Wythoff 's game with move vector {(1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 1)} generates the Sierpinski sponge.…”