From:mverona@cs.umd.edu (Mary Verona)Newsgroups:rec.puzzles,sci.mathSubject:Wythoff's gameSummary:Need help for a three-dimensional extension of Wythoff's GameKeywords:Wythoff's Game, Wythoff's Nim, Wythoff pairs, Fibonacci sequences,Date:9 Jul 92 01:19:59 GMTOrganization:U of Maryland, Dept. of Computer Science, Coll. Pk., MD 20742

I have a student who is working on a three-dimensional extension of Wythoff's Game. We would appreciate any help that may be available. Two-dimensional Wythoff's Game is a combinatorial game played by two players using two piles of objects. On each turn, a player may remove some or all of the objects from one pile or the same number of objects from both piles. The goal of the fame is for a player to remove all of the objects on his or her turn. The study of Wythoff's Game centers around the study of Wythoff's pairs, which are positions in the Game where a player is guaranteed the win if he or she has just moved to that position. A Wythoff pair is an ordered pair (a,b) where a and b represent the number of objects in the respective piles. It turns out that the Wythoff pairs have several interesting Fibonacci and golden ratio properties. My student has attempted to move the game into three dimensions, using three piles. He hopes to find similar properties in the new variation. If you have any help or would like to see the Wythoff triples, please write @schien.cs.uah.edu. ordered pair (a,b) where a and b represent the number of objects in the respective piles. It turns out that these pairs have several interesting properties involving the Fibonaaci sequence and the golden ratio. w win if he or she has just moved to that position. A Wyt hoff pair is an