Wythoff'snim and Fibonacci Representations

Our aim in what follows is to show how Fibonacci representations play a role in determining winning moves for Wythoffs Nim [1, 2] very analogous to the role of binary representations in Bouton's Nim [3 ] . The particulars of these two games can be found in the preceding references, but for the convenience of the reader, we briefly recount the rules of play. In Bouton's Nim (usually referred to simply as nim) two players alternate picking up from a given collection of piles of counters (such as stones or coins). In his turn, a player must pick up at least one counter, so is never allowed a "pass." All counters picked up in one turn must come from a single pile, although the selection of pile can be changed from turn to turn. The number of counters picked up is constrained above by the size of the selected pile, but is otherwise an open choice on each move. The player who makes the last move (picking up the last counter) is declared the winner. In Wythoffs Nim, only two piles are involved. On each turn a player may move as in Bouton's Nim, but has the added option of picking up from both piles, provided he picks up an equal number of counters from the two piles. As in Bouton's Nim, the winner is the player who makes the last move. As has been known since Wythoffs original paper [1 ] , the strategy for Wythoffs Nim consists of always leaving the opponent one of a sequence of pairs (1,2), (3,5), (4,7) , -