How to Beat Your Wythoff Games' Opponent on Three Fronts

TLDR: In the Wythoff game, a player may remove any positive number of tokens from a single pile, or he may take from both piles, provided that I k 11 0, is an N-position for every a; the Next player moves to (0, 0) and wins as discussed by the authors.

Abstract: 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.