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## 'On the Edge' printed from http://nrich.maths.org/

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.

Adam's square looks like this:
Arthur sent us these great pictures
for four by four and five by five squares. Thank you
Arthur!

Jenny made a prediction:
``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."