Counting, Symmetries and Equivalence Classes of Sudoku Grids

Fernanda Pereira

arXiv:2607.20669v2 Primary: math.GM

Abstract

Sudoku is a widely popular puzzle whose complete grids have been enumerated computationally: there are approximately $6.67 \times 10^{21}$ 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.

Disclosure

“24 FERNANDA PEREIRA Use of artificial intelligence tools While preparing this work, the author used the artificial intelligence as- sistant Claude (Claude Sonnet 5 model, accessed via claude.ai) for language editing and translation, for assistance in deriving and developing some of the mathematical arguments, and for writing the Python scripts used in the computational verifications”
PDF page 24

Classification: Proof ideas or individual proof-step assistance
Multiplier: 8
Verified: 2026-08-23T19:04:49.076070Z

Structural counts

Pages 24 pdf

Theorems 3 source

Lemmas 2 source

Propositions 3 source

Corollaries 0 source

Definitions 3 source

Displayed equations 31 source

Bibliography entries 10 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file sudoku.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.