Seckendorf states that any positive integer is uniquely represented as the sum of non-consecutive Fibonacci numbers indexed by $F_1 = 1, F_2 = 2, F_{n+1} = F_n + F_{n-1}$ Proved to be disassembled. Motivated by this result, Baird, Epstein, Flint, and Miller defined a two-player Zeckendorff game in which the two players take turns acting on multiple sets of Fibonacci numbers totaling $N$. The game ends when there are no more possible moves, and the last player to play a hand wins. In particular, I studied random game settings. The game proceeds by randomly selecting the available hands uniformly, and as we enter $N \to \infty$ we infer that the distribution of random game lengths converges to a Gaussian distribution.

    Prove that a certain sum of move counts is constant and find a lower bound on the shortest number of games for input $N$ containing Catalan digits. The work of Baird et al. and Cuzensa et al. For a given input $N$, we determined how to achieve the shortest and longest possible Seckendorf games, respectively. For any given input $N$, we establish that the range of possible game lengths constitutes an interval of natural numbers. Maximum game length can be achieved.

    We further explore the probabilistic aspects of random Seckendorf games. Examine two probability measures on the space of all Seckendorf games for input $N$. Under both measures with a limit of $N \to \infty$ both players have a probability of $1/2$ winning. We also find a natural division of the collection of all Seckendorf games for fixed input $N$. In this case we observe a weak convergence to a Gaussian distribution in the limit $N \to \infty$ . Finish work with many open issues.

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