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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…

Walnut is a software that using automata can prove theorems in combinatorics on words about automatic sequences. We are able to apply this software to both prove new results as well as reprove some old results on avoiding squares and cubes…

Let $\alpha = (1+\sqrt{5})/2$ and define the lower and upper Wythoff sequences by $a_i = \lfloor i \alpha \rfloor$, $b_i = \lfloor i \alpha^2 \rfloor$ for $i \geq 1$. In a recent interesting paper, Kawsumarng et al. proved a number of…

We study the problem whether there exist variants of {\sc Wythoff}'s game whose $\P$-positions, except for a finite number, are obtained from those of {\sc Wythoff}'s game by adding a constant $k$ to each $\P$-position. We solve this…

We modify Wythoff's game by allowing an additional move, which we call a "split", and show how the $P$-positions are coded by the Tribonacci word. We analyze the table of letter positions of arbitrary $k$-bonacci words and find a…

We introduce a variant of Wythoff's Game that we call $m$-Modular Wythoff's Game. In the original Wythoff's Game, players can take a positive number of tokens from one pile, or they can take a positive number of tokens from both piles if…

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…

Wythoff's Game is a game for two players playing alternately on two stacks of tiles. On her turn, a player can either remove a positive number of tiles from one stack, or remove an equal positive number of tiles from both stacks. The last…

We show that the winning positions of a certain type of two-player game form interesting patterns which often defy analysis, yet can be computed by a cellular automaton. The game, known as {\em Blocking Wythoff Nim}, consists of moving a…

We prove three conjectures of Fraenkel and Ho regarding two classes of variants of Wythoff's game. The two classes of variants of Wythoff's game feature restrictions of the diagonal moves. Each conjecture states that the Sprague-Grundy…

We begin by reviewing and proving the basic facts of combinatorial game theory. We then consider scoring games (also known as Milnor games or positional games), focusing on the "fixed-length" games for which all sequences of play terminate…

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 :=…

The 2-player impartial game of Wythoff Nim is played on two piles of tokens. A move consists in removing any number of tokens from precisely one of the piles or the same number of tokens from both piles. The winner is the player who removes…

We introduce a restriction of Wythoff's game, which we call F-Wythoff, in which the integer ratio of entries must not change if an equal number of tokens are removed from both piles. We show that P-positions of F-Wythoff are exactly those…

Walnut is a software package that implements a mechanical decision procedure for deciding certain combinatorial properties of some special words referred to as automatic words or automatic sequences. Walnut is written in Java and is open…

This paper describes Wythoff's game with a pass, which is 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…

We propose a new interpretation of the strange phenomena that some authors have observed about the Wald game. This interpretation is possible thanks to the new language of \emph{loadings} that Morrison and the author have introduced in a…

We present two variants of Wythoff's game. The first game is a restriction of Wythoff's game in which removing tokens from the smaller pile is not allowed if the two entries are not equal. The second game is an extension of Wythoff's game…

Wythoff's Nim is a variant of 2-pile Nim in which players are allowed to take any positive number of stones from pile 1, or any positive number of stones from pile 2, or the same positive number from both piles. The player who makes the…

In an investigation of the applications of Combinatorial Game Theory to chess, we construct novel mutual Zugzwang positions, explain an otherwise mysterious pawn endgame from "A Guide to Chess Endings" (Euwe and Hooper), show positions…