Winning and Losing Positions and Some Grundy Values for the Game of Exact-2-Fibonacci NIM
Abstract
Fibonacci NIM is an impartial combinatorial game, usually played with a single pile of stones, Its analysis involves the Fibonacci numbers and the Zeckendorf representation of a natural number. In this paper we will deal a study and a mathematical analysis of the case of 2-pile and 3-pile with the condition that the stones are deleted from exactly two piles, and we find Grundy values for two piles (n1, n2; r1, r2) and the winning strategy which amounts to understanding positions of Grundy value 0, and we will calculate all the positions of Grundy value {0,1,2,3}, and use Fibonacci words to calculate all Grundy values for any position in which r1 is {1,2,3,4,5}.
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