Exact Maxima for the Three Open Stalemate Problems of Marco Ripà: A Complete Combinatorial and Computational Resolution under FIDE Laws

We resolve the three open problems posed by Marco Ripà concerning maximal material imbalances attainable in legal chess positions that end in stalemate under the FIDE Laws of Chess (including the current rules on dead positions, fivefold repetition and the 75-move rule). Writing S for the stalemated side and F for the free side, the three quantities are the maximum of Score(S) minus Score(F), the maximum of Score(S), and the maximum of Score(S) plus Score(F) when S is stalemated. We prove that the constructive lower bounds discovered by Ripà—namely 81, 91 and 168—are in fact the exact theoretical maxima. The proofs rest on an exact algebraic identity for the material score of a stalemated army containing at least one queen, a geometric enclosure lemma for queens, an analysis of the unblockable frontier of a stalemated set, exhaustive enumeration of all queen clusters of size at most 9 whose closed king-neighbourhood has cardinality at most 16, and a file-clearing inequality that governs the necessary captures when both sides promote. All case distinctions are verified by a self-contained Python program that performs a complete combinatorial search in approximately fifteen seconds on a standard CPU. The computer certificate confirms that no configuration with eight or nine queens can be stalemated, that the maximum score of a stalemated army is exactly 91, that the maximum difference is exactly 81, and that the maximum total material in any stalemate is exactly 168.