Box Canonicalization of a Floor

A band of 3 rows from a Sudoku solution grid can be modified to its minimum lexicographic equivalent.

For a Sudoku grid with only givens a different algorithm is used, to account for missing digits.

Steps to find all 108 starting permutations

• Try each of the 3 boxes as box 1. (3 permutations)
• Permute the 3 rows. (6 permutations)
• Permute the 3 columns in box 1. (6 permutations)

Steps to normalize a starting permutation

• Relabel the numbers so that box 1 reads:

123
456
789

• Sort box 2 and 3 so that box 2 contains 4 in the top row.
• Sort the columns in box 2 so that r1c4=4, r1c5 < r1c6
• Sort the columns in box 3 so that r1c7 < r1c8 < r1c9.

Notes

There are a total of 36288 different configurations for a normalized floor, 416 of which are unique.

Compare results

• The numbers to compare can be found in cells a-q below:

1 2 3 | 4 a b | c d e
4 5 6 | f g h | i j k
7 8 9 | l m n | o p q

• Of the 108 starting permutations, choose the one with the lowest numerical value
• For all possible configurations of the 27 numbers in a band, there are only 416 essentially different ones.