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 multiplicative composition of MuM positions is governed by modular multiplication, in contrast to classical Nim where composition is governed by XOR (nim-sum). 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 \(\mathbb {F}(p^n)\) , 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 an integer heap model with synonym-move exclusion 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.
