6 problems

Longest-game conjecture. The longest game on any $n$ is achieved by applying splitting moves whenever possible: first adding $1$'s, then splitting from smallest to largest, and fin…

Gaussian-limit conjecture. As $n$ tends to infinity, the number of moves in a random game converges to a Gaussian distribution.

Baird–Epstein–Flint–Miller Gaussianity conjecture. As $N\to\infty$, the distribution of the game length converges to a Gaussian distribution, with expectation and variance approxim…

Let $n$ be a positive integer, and consider the Split Smallest game, the deterministic Zeckendorf game that prioritizes splitting moves from smallest to largest, then adding 1's, a…

Gaussian conjecture. The number of splitting moves in a random game converges to a Gaussian distribution, with mean and variance approximately $0.215n$.

Linear-growth conjecture. For Split Smallest, the number of moves grows linearly with $n$, with the asymptotic constant appearing numerically to be the square of the golden mean. F…