  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 $n \ 1$'s and take turns applying rules
inspired by the Fibonacci recurrence, $F_{n+1} = F_n + F_{n-1}$, 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, allowing us to construct
a variation on the Zeckendorf Game, the Fibonacci Quilt Game. While some
properties of the Fibonaccis are inherited by this sequence, the nature of its
recurrence leads to others, such as Zeckendorf's theorem, no longer holding; it
is thus of interest to investigate the generalization of the game in this
setting to see which behaviors persist. We prove, similar to the original game,
that this game also always terminates in a legal decomposition, give a lower
bound on game lengths, show that depending on strategies the length of the game
can vary and either player could win, and give a conjecture on the length of a
random game.
