Related papers: Two variants of Wythoff's game preserving its P-po…
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…
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 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…
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…
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…
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…
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 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…
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 study impartial take away games on 2 unordered piles of finite nonnegative numbers of tokens $(x,y)$. Two players alternate in removing at least one and at most all tokens from the respective piles, according to certain rules, and the…
We study a variation of the combinatorial game of 2-pile Nim. Move as in 2-pile Nim but with the following constraint: Suppose the previous player has just removed say $x>0$ tokens from the shorter pile (either pile in case they have the…
Given $k\ge 3$ heaps of tokens. The moves of the 2-player game introduced here are to either take a positive number of tokens from at most $k-1$ heaps, or to remove the {\sl same} positive number of tokens from all the $k$ heaps. We analyse…
Given $n$ piles of tokens and a positive integer $k \leq n$, we study the following two impartial combinatorial games Nim$^1_{n, \leq k}$ and Nim$^1_{n, =k}$. In the first (resp. second) game, a player, by one move, chooses at least $1$ and…
In this paper we study a family of 2-pile Take Away games, that we denote by Generalized Diagonal Wythoff Nim (GDWN). The story begins with 2-pile Nim whose sets of options and $P$-positions are $\{\{0,t\}\mid t\in \N\}$ and $\{(t,t)\mid…
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 study 2-player impartial games of the form take-away which produce P-positions (second player winning positions) corresponding to complementary Beatty sequences, given by the continued fractions (1;k,1,k,1,...) and (k+1;k,1,k,1,...). Our…
Here, we present a variant of the sliding coins game. Two coins are placed on distinct squares of a semi-infinite linear board with squares numbered $0, 1, 2, dots, $. Two players take turns and move a coin to a lower unoccupied square.…
On a semi-infinite strip of squares rightward numbered $0, 1, 2, \ldots$ with at most one coin in each square, in Welter's game, two players alternately move a coin to an empty square on its left. Jumping over other coins is legal. The…