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Sudoku determinant. Halp.

  1. May 29, 2013 #1
    1. The problem statement, all variables and given/known data
    A while ago someone posted this problem:
    -----------------------------------------------------------------------------------
    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.
    -------------------------------------------------------------------------------

    2. Relevant equations

    3. 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.
     
  2. jcsd
  3. May 29, 2013 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    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.
     
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