Given a $10×10$ grid with $9$ red blocks and $91$ white blocks, in each step we color one red block black and after that one white block red until there are $91$ black blocks and $9$ red blocks. Prove that there exits a step in which a black block shares an edge with a white block.
I ran a few trials and couldn't get past the fifth step without adjacency. But I don't know how to build a proof.