Nest graphs and minimal complete symmetry groups for magic Sudoku variants

Felgenhauer and Jarvis famously showed in [2], although it was first mentioned earlier, in [7], that there are 6,670,903,752,021,072,936,960 possible completed Sudoku boards. In a later paper, Jarvis and Russell [8] used a Sudoku symmetry group of size 3, 359, 232 · 9! = 1, 218, 998, 108, 160 and Burnside’s Lemma to show that there are 5,472,730,538 essentially different Sudoku boards. Both of these results required extensive use of computers as magnitude of the numbers makes non-computer exploration of these problems prohibitively difficult. The ongoing goal of this project is to find and implement methods to attack these and similar questions without the aid of a computer. One step in this direction is to reduce the size of the symmetry group with purely algebraic, non-computer methods. The strategy of [1], applied to the analogous symmetry group for a 4 × 4 Sudoku variation known as Shidoku, was to partition the set of Shidoku boards into so-called H4-nests and S4-nests and then use the interplay between the physical and relabeling symmetries to find certain subgroups of G4 that were both complete and minimal. A symmetry group is complete if its action partitions the set of Shidoku boards into the two possible orbits, and minimal if no group of smaller size would do the same. In [4], Lorch and Weld investigated a 9 × 9 variation of Sudoku called modular-magic Sudoku that has sufficiently restrictive internal structure to allow for non-computer investigation. In this paper we will apply the techniques from [1] to find a minimal complete symmetry group for the modularmagic Sudoku variation studied in [4], as well as for another Sudoku variation that we will call semi-magic Sudoku.