Coloring 4 by 3 square such that every unit square has an even number of squares it shares a vertex with of the same color (including itself).

Question

Coloring 4 by 3 square such that every unit square has an even number of squares it shares a vertex with of the same color (including itself).

I don't think this is possible, I have done a fair bit of case bashing to this effect but I was wondering if there was a more elegant way to approach it? Thanks so much.

Note seems closely related to the 3 by 3 case but the same nice solution doesn't work. Is there a construction. Thanks so much

Answer 1

Maybe I'm not understanding the problem properly, but what if you coloured the squares with colours 1 and 2 like so? $$\begin{array}{ccc} 1 & 1 & 1\\ 1 & 1 & 1\\ 2 & 2 & 2\\ 2 & 2 & 2 \end{array}$$