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arXiv · 2608.06162

Do We Really Need to Read the Input? An Optimality Proof for Stone Game III

Andrew Au
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Abstract

Stone Game III admits a standard backward dynamic program using $O(n)$ time and $O(1)$ auxiliary space. The upper bound is immediate, but its optimality raises a deceptively simple question: must a correct algorithm really inspect a linear number of input values? For the original problem, an all-zero instance gives a short indistinguishability proof that every position must be inspected. This argument appears to depend strongly on the possibility of a tie. We show that it does not. Even under the promise that every input has a winner, an adversary can force any deterministic algorithm to make $\Omega(n)$ inspections by combining modular move control with indistinguishable input completions. We also extend the argument to positive but unbounded values, obtaining the same linear lower bound without zeros or ties. Together these results establish the asymptotic optimality of the standard $O(n)$-time, $O(1)$-space solution in several increasingly restrictive variants.

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Andrew Au. 2026-08-06. Do We Really Need to Read the Input? An Optimality Proof for Stone Game III. https://arxiv.org/abs/2608.06162

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