My last post was a little tiling puzzle: you can read it here. In this post I want to quickly give the solution.
Represent a red tile by \(1\) and a blue tile by \(-1\); and think of the square in coordinate \((a,b)\) as the monomial \(x^ay^b\). Then the question is equivalent to asking when the polynomial \(p_{N,M}(x,y)=\sum_{a=1}^N\sum_{b=1}^Mx^ay^b\) is in the ideal generated by \((1-x+x^2, 1-y+y^2)\). These are cyclotomic polynomials for sixth roots of unity, so one can test whether \(p_{N,M}(x,y)\) is in this ideal by evaluating it at \((\zeta, \zeta)\), where \(\zeta\) is a primitive sixth root of unity. Now it’s an exercise to check that \(p_{N,M}(\zeta, \zeta)=0\) if and only if one of \(N,M\) is divisible by 6!