Zeckendorf proved that every positive integer has a unique representation as
a sum of non-consecutive Fibonacci numbers. Once this has been shown, it's
natural to ask how many summands are needed. Using a continued fraction
approach, Lekkerkerker proved that the average number of such summands needed
for integers in $[F_n, F_{n+1})$ is $n / (\varphi^2 + 1) + O(1)$, where
$\varphi = \frac{1+\sqrt{5}}2$ is the golden mean. Surprisingly, no one appears
to have investigated the distribution of the number of summands; our main
result is that this converges to a Gaussian as $n\to\infty$.