How many ways to colorize the table Each cell of a square table of 3 × 3 can be painted black, white or red. How many different colorings of this table? If we do not consider the turnings of the board, we get $3^9$ variants. How many variants of coloring will be, if you consider the possibility of turning the board? I can sort through all the options for a 2 × 2 table and two colors (16 variants without rotation and 6 variants with turns), but for 3 × 3 variants too much.

The formula is Burnside's Lemma. You take a group, in this case the group of rotations of a square: $0, 90, 180, 270$ degree rotations, and look at which colourings are fixed by each rotation. The formula is

> $$ \frac{1}{\text{size of group}}\left( \text{colourings fixed by the first group element} + \text{colourings fixed by the second group element} + \cdots \right) $$

In this case

* $3^9$ colourings are fixed by a $0$ degree rotation

* $3^3$ colourings are fixed by a $90$ or $270$ degree rotation (such colourings are determined by how the top-left, top-centre and centre squares are coloured)

* $3^5$ colourings are fixed by a $180$ degree rotation (determined by the top three squares and the centre-left and centre squares)




Applying Burnside's Lemma, there are

$$ \frac14 \left( 3^9 + 2 \cdot 3^3 + 3^5 \right) = 4995 $$

distinct colourings.