Optimizing Sudoku Determinants: Finding the Minimum and Maximum Values

[alexeih]

Homework Statement

A while ago someone posted this problem:
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Problem 1 (given after a discussion of determinants in week 3/4 of the course):
Consider a 9x9 matrix A. We say that A is a Sudoku matrix if it's the valid solution to a Sudoku puzzle. That is if,
1) Every row and every column is a permutation of {1,2,3,4,5,6,7,8,9}.
2) If we write it in block form:
A=
A1 A2 A3
A4 A5 A6
A7 A8 A9

where Ai is a 3x3 matrix, then every Ai has elements {1,2,3,4,5,6,7,8,9}.
Now the problem is:
a) Find a Sudoku matrix with determinant 0.
b) Does there exist a Sudoku matrix with determinant 1. If not then determine the least positive number that a Sudoku matrix can have as a determinant.
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Homework Equations

The Attempt at a Solution

I've managed to get a) and a lower bound of 405 on b), but showing *the* lower bound is eluding me. I wrote a mini generator in matlab, so that when I do a relatively simple permutation, like switching 1 and 2 in a singular matrix it generates large determinants, so my postulate is that it's something like 5*3^9, but I'm tearing my hair out here.

 

[mfb]

It would be interesting how you got that lower bound.

In addition, did you calculate the determinant of some random sudokus?

Formulas for determinants of block matrices could be interesting.