Adam of Moorside School sent us in his description of a possible painting for the three by three square:

The outside of the square should be green. The rest of one of the corner squares should be yellow. The rest of another corner square should be blue. The other two corner squares should have one side blue and the other side yellow. Then two of the squares in between the corner squares should have two touching sides yellow and one side blue. The other two squares in between the corner squares should have two touching sides blue and one side yellow. The middle square should be half yellow, half blue.

I noticed that in an $n \times n$ square there are a total on $n^2$ small squares, and therefore a total of $4 n^2$ edges to be coloured. Each colour takes up $4 n$ of these edges, so it should be possible to colour an $n \times n$ square with $n$ different colours. To do this, we need to arrange for all the colours to have four cornerpieces and $4 (n-2)$ edge pieces."