2018/09/13 by Baird-Smith, Paul, Epstein, Alyssa, Flint, Kristen +1 · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1809.04881
Zeckendorf proved that every positive integer n can be written uniquely as the sum of non-adjacent Fibonacci numbers. We use this to create a two-player game. Given a fixed integer n and an initial decomposition of n = n F1, the two players alternate by using moves related to the recurrence relation Fn+1 = Fn + Fn-1, and whoever moves last wins. The game always terminates in the Zeckendorf decomposition, though depending on the choice of moves the length of the game and the winner can vary. We find upper and lower bounds on the number of moves possible. The upper bound is on the order of nlog n, and the lower bound is sharp at n-Z(n) moves, where Z(n) is the number of terms in the Zeckendorf decomposition of n. Notably, Player 2 has the winning strategy for all n > 2; interestingly, however, the proof is non-constructive.