2026/07/22 by Fernanda Pereira
Mathematics · #math.GM
Sudoku is a widely popular puzzle whose complete grids have been enumerated computationally: there are approximately 6.67 × 1021 of them and, up to symmetry and renaming of digits, 5,472,730,538 essentially different ones. The classical enumeration reduces the count to 44 equivalence classes of the first band through a chain of ad hoc reductions, leaving the number 44 without any apparent structural explanation. We present an alternative derivation of these 44 classes, in which they arise as isomorphism classes of unordered triples (multisets) of column partitions under relabeling, a single invariant that replaces the original chain of reductions. This invariant makes it possible to apply Burnside's Lemma by hand: we recover 44 through a closed derivation requiring no computational enumeration.