Sorry, it turns that was misunderstanding from my part. It turns out we cannot solve a board row by row without revisiting the previous rows. The board below is a counterexample:

  101
  010
  100

The first and second rows are solved. However we cannot solve the last row without re-doing the first and second rows. The two solutions of this board shows this:

  solution #1:
  110
  100
  000

  solution #2:
  110
  110
  000

Actually, seeing my mistake made me challenge my assumption that a singular matrix over the reals might not be over integers modulo 2. This is likely wrong too. I don't know much about abstract algebra (and I am not a mathematician). Wikipedia (https://en.wikipedia.org/wiki/Determinant#Square_matrices_ov...) states "the reduction modulo m of the determinant of such a matrix is equal to the determinant of the matrix reduced modulo m."

The move matrix M for all boards of size 3n + 2 appears to be singular. This means these boards may have no solution or a large number of solutions.