1. Wythoff Games. Let a be a positive integer. Given two piles of tokens, two players move alternately. The moves are of two types: a player may remove any positive number of tokens from a single pile, or he may take from both piles, say k (> 0) from one and 1 (> 0) from the other, provided that I k 11 0, is an N-position for every a; the Next player moves to (0, 0) and wins. For a = 2, the position (1, 3) is a P-position: if Next moves to (0, 3), (0,2) or (0, 1), then Previous, using a move of the first type, moves to (0, 0) and wins. If Next moves to (1, 2) or to (1, 1), then Previous, using a move of the second type, can again move to (0, 0). The set of all P-positions is denoted by P, and the set of all N-positions by N.

How to Beat Your Wythoff Games' Opponent on Three Fronts

Let θ = θ(k) be the positive root of θ 2 + (k-2)θ-k = 0. Let f(n) = [(n + l)θ]-[nθ] for positive integers n, where [x] denotes the greatest integer in x. Then the elements of the infinite sequence (f(l), f(2), f(3),…) can be rapidly generated from the finite sequence (f(l), f(2),…,f(k)) by means of certain shift operators. For k = 1 we can generate (the characteristic function of) the sequence [n θ] itself in this manner.

Beatty Sequences, Continued Fractions, and Certain Shift Operators

Roughly speaking, the family of combinatorial games consists of two-player games with perfect information (no hidden information as in some card games), no chance moves (no dice) and outcome restricted to (lose, win), (tie, tie) and (draw, draw) for the two players who move alternately. Tie is an end position such as in tic-tac-toe, where no player wins, whereas draw is a dynamic tie: any position from which a player has a nonlosing move, but cannot force a win. Both the easy game of Nim and the seemingly difficult chess are examples of combinatorial games. And so is go. The shorter terminology game, games is used below to designate combinatorial games.

Combinatorial games: Selected bibliography with a succinct gourmet introduction.

Let α be a positive irrational real number. (Without loss of generality assume 0 < α < 1.) The characteristic sequence of α is f(α) =f1f2 ···, where fn = [(n + 1)α] - [nα]. We make some observations on the various descriptions of the characteristic sequence of α which have appeared in the literature. We then refine one of these descriptions in order to obtain a very simple derivation of an arithmetic expression for [nα] which appears in A. S. Fraenkel, J. Levitt, and M. Shimshoni [17]. Some concluding remarks give conditions on n which are equivalent to fn = 1.

I. Borosh

Papers

A generalization of Wythoff's game