Optimal Pile Balancing In Uniform Random Depletion Systems

We resolve a conjecture posed by Gutiérrez and Subercaseaux (arXiv:2403.03330) concerning a discrete stochastic process on multisets. The process begins with N indistinguishable rocks distributed across K piles. At each time step, a non-empty pile is selected uniformly at random, and one rock is removed. The process terminates when exactly one pile remains non-empty. The conjecture asserts that the expected duration of the process is maximized when the initial configuration is balanced, that is, when the pile sizes differ by at most one. In this paper, we prove the conjecture for all values of N and K, providing a full resolution. The proof proceeds by showing that local rebalancing operations (transferring a rock from a larger to a smaller pile) strictly increase the expected process duration. This yields a global maximum at balanced configurations via a finite descent argument over the configuration space. We also present entropy-based and symmetry-theoretic interpretations of the result, reinforcing the intuition that balanced initial states maximize persistence under uniform stochastic erosion. Empirical results and exact recurrence computations are included to illustrate and verify the theoretical findings. Also, hopefully, we win 100$.