Combinatorics ( colouring a checkerboard) Question:
If we identify colourings related by rotations of a checkerboard, how many ways can you paint it if it has 2Tx2T fields with m colours?
My attempt:
I'm struggling to begin this. The part of the question I am confused with is the "if we identify colourings related by rotations of a checkerboard".

A solution that does not use Burnside's lemma explicitly:
There are $m^{4t^2}$ possible boards, and naively we must divide this by $4$. However:

*

*$m^{2t^2}$ boards are equivalent under a $180^\circ$ turn, so we must "boost" the count by $m^{2t^2}$ so that these boards are counted once (not $0.5$ times)

*$m^{t^2}$ boards are equivalent under a $90^\circ$ turn, so we must "boost" the count by $2m^{t^2}$ so that these boards are counted once (not $0.5$ times; before the first bullet point above they are counted $0.25$ times)

Thus the final answer is, writing $z=m^{t^2}$ (the number of ways to colour a fixed $t×t$ grid with $m$ colours),
$$\frac{z^4+z^2+2z}4$$
