Modular magic sudoku

Abstract

A modular magic sudoku solution is a solution to a sudoku puzzle with symbols in {0, 1,. .. , 8} such that rows, columns, and diagonals of each subsquare add to 0 mod 9. We count these sudoku solutions by using the action of a suitable symmetry group and we also describe maximal mutually orthogonal families.

Nest graphs and minimal complete symmetry groups for magic Sudoku variants

“…In [4] it is shown that the set of modular-magic boards breaks into two orbits under the action of G mm , with representatives shown in Figure 2 Every 3 × 3 block in a modular-magic Sudoku board has two minidiagonals, one of which must be from the set {0, 3, 6}. Therefore each modular-magic Sudoku board has exactly three blocks with center entry 0, three with center entry 3, and three with center entry 6.…”

“…As mentioned in the proof of Lemma 1, the set of modular-magic boards is a union of two G mm -orbits. Observe that the three H mm -nests represented by [1,1], [2,2], and [7,7] lie in the G mm -orbit containing the left board of Figure 2, which has size 4608 according to [4]. Meanwhile, the remaining six H mm -nests lie in the same G mm -orbit as the right-hand board of Figure 2, which has size 27648 by [4].…”

“…Observe that the three H mm -nests represented by [1,1], [2,2], and [7,7] lie in the G mm -orbit containing the left board of Figure 2, which has size 4608 according to [4]. Meanwhile, the remaining six H mm -nests lie in the same G mm -orbit as the right-hand board of Figure 2, which has size 27648 by [4]. This tells us that the three H mm -nests represented by [1,1], [2,2], and [7,7] have size 4608/3 = 1536 each while the remaining six H mm -nests are each of size 27, 648/6 = 4608.…”

Magic Cayley-Sudoku Tables

“…Lorch and Weld [3] defined a modular magic sudoku table as an ordinary sudoku table (with 0 in place of the usual 9) in which the row, column, diagonal, and antidiagonal sums in each 3 × 3 block in the table are zero mod 9. "Magic" refers, of course, to magic Latin squares which have a rich history dating to ancient times.…”

“…. Thus, the label [0, 1, 2] is interpreted as the elements 0, 1, and 2 listed vertically, one per row in the indicated order, and [0, 3,6] is interpreted as the elements 0, 3, and 6 listed horizontally, one per column in the indicated order, and so forth.…”