The Fibonacci Quilt Game
Steven J. Miller, Alexandra Newlon

TL;DR
This paper introduces the Fibonacci Quilt Game, a generalization of the Zeckendorf Game based on the Fibonacci Quilt sequence, analyzing its properties, termination, and strategic variations.
Contribution
It extends the Zeckendorf Game to the Fibonacci Quilt sequence, proving termination, analyzing game length bounds, and exploring strategic outcomes.
Findings
The game always terminates in a legal decomposition.
Game length can vary depending on strategies.
Both players can win with optimal play.
Abstract
Zeckendorf proved that every positive integer can be expressed as the sum of non-consecutive Fibonacci numbers. This theorem inspired a beautiful game, the Zeckendorf Game. Two players begin with 's and take turns applying rules inspired by the Fibonacci recurrence, , until a decomposition without consecutive terms is reached; whoever makes the last move wins. We look at a game resulting from a generalization of the Fibonacci numbers, the Fibonacci Quilt sequence. These arise from the two-dimensional geometric property of tiling the plane through the Fibonacci spiral. Beginning with 1 in the center, we place integers in the squares of the spiral such that each square contains the smallest positive integer that does not have a decomposition as the sum of previous terms that do not share a wall. This sequence eventually follows two recurrence relations,…
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