The linear system for Sudoku and a fractional completion threshold

We study a system of linear equations associated with Sudoku latin squares. The coefficient matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math> of the normal system has various symmetries arising from Sudoku. From this, we find the eigenvalues and eigenvectors of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math> , and compute a generalized inverse. Then, using linear perturbation methods, we obtain a fractional completion guarantee for sufficiently large and sparse rectangular-box Sudoku puzzles.