# Establish necessary and sufficient condition for specific matrix color configuration

Given a 2n by 2n matrix of cells, if each row and each column has exactly n cells of red color and remaining n cells of blue color, prove that for each row with reds in particular positions, there is another row with only blues in those positions.

Conversely, if rows cannot be paired as described above, prove that the matrix cannot be colored with each row and each column containing equal number of reds and blues.

NOTE: What I have asked above is to prove a conjecture that is a figment of my thought process which I used as part of solving a different puzzle.

Your conjecture is false. Below, none of rows $1$ through $4$ have a complement.
or $$\begin{matrix} 1&1&0&0&1&0\\ 0&1&1&0&0&1\\ 0&0&1&1&1&0\\ 1&0&0&1&0&1\\ 1&0&1&0&1&0\\ 0&1&0&1&0&1 \end{matrix}$$
• Nice! I looked for $4 \times 4$ counterexamples, became mostly convinced that there were none, but that for larger sizes there might be...but didn't get as far as constructing one. Commented May 16, 2018 at 17:46