Nest graphs and minimal complete symmetry groups for magic Sudoku variants

TL;DR

This paper extends symmetry group analysis to various Sudoku variants, identifying minimal complete groups and revealing that the full Sudoku symmetry group is inherently minimal and complete.

Contribution

It applies nest graph techniques to find minimal complete symmetry groups for magic Sudoku variants, including semi-magic Sudoku, and shows the full group is already minimal and complete.

Findings

Full Sudoku symmetry group is already minimal and complete.

Identified minimal complete symmetry groups for magic and semi-magic Sudoku.

Extended nest graph methods to new Sudoku variants.

Abstract

A symmetry group for Sudoku is complete if its action partitions the set of Sudoku boards into all possible orbits, and minimal if no group of smaller size would do the same. Previously, for a 4 x 4 Sudoku variation known as Shidoku, the authors used an analogous symmetry group to partition the set of Shidoku boards into so-called "nests" and then use the interplay between the physical and relabeling symmetries to find certain subgroups that were both complete and minimal. In this paper these same techniques are applied to find a minimal complete symmetry group for the modular magic Sudoku variation, as well as for another Sudoku variation called semi-magic Sudoku. The paper concludes with a simple computation which leads to the non-obvious fact that the full Sudoku symmetry group is, in fact, already minimal and complete.