标题： 乘法模态尼姆（MuM） 显示英文标题

标题： Multiplicative Modular Nim (MuM)

摘要： 我们引入了乘法模Nim（MuM），这是Nim的一种变体，其中传统的nim和被堆大小的模m乘法所取代。我们为这个游戏建立了完整的理论，从针对素数模数的直接的Bouton式分析开始。我们的核心结果是Sprague-Grundy定理的一个类比，其中我们通过乘法mex递归为每个位置定义了一个博弈论值，即mumber。我们证明这些mumber等价于堆积模m，并表明对于游戏的析取和，它们通过模乘法组合，而不是经典nimbers的异或和。对于合数模数，我们证明MuM通过中国剩余定理分解为其素数幂因子对应的独立子游戏。我们还将游戏扩展到有限域F(pn)，这是出于使AES S盒代数更易理解的教学需求。我们证明在这个领域中，一个健全的游戏需要一个规范堆模型来解决从整数堆到域元素的多对一映射。据我们所知，这是对Nim的乘法模变体及其扩展到完整、非加性组合博弈代数的首次系统分析。 显示英文摘要

摘要： We introduce Multiplicative Modular Nim (MuM), a variant of Nim in which the traditional nim-sum is replaced by heap-size multiplication modulo m. We establish a complete theory for this game, beginning with a direct, Bouton-style analysis for prime moduli. Our central result is an analogue of the Sprague-Grundy theorem, where we define a game-theoretic value, the mumber, for each position via a multiplicative mex recursion. We prove that these mumbers are equivalent to the heap-product modulo m, and show that for disjunctive sums of games, they combine via modular multiplication in contrast to the XOR-sum of classical nimbers. For composite moduli, we show that MuM decomposes via the Chinese Remainder Theorem into independent subgames corresponding to its prime-power factors. We extend the game to finite fields F(pn), motivated by the pedagogical need to make the algebra of the AES S-box more accessible. We demonstrate that a sound game in this domain requires a Canonical Heap Model to resolve the many-to-one mapping from integer heaps to field elements. To our knowledge, this is the first systematic analysis of a multiplicative modular variant of Nim and its extension into a complete, non-additive combinatorial game algebra.