The following problem was given as a homework problem, so I am not necessarily asking for a full solution, but rather a good hint on where to start.
A chess board, where some of the $64$ cells contain a piece, has the property that every row and every column contains $3$ pieces exactly. I need to show that one can remove some pieces, such that the resulting matrix has exactly one piece in every row and every column.
I feel like this must have an easy solution, but I have not been able to guess it. To give a little bit of context, this problem was given in relation to totally unimodular matrices and integrality of polyhedra.