Trish and James sent us in their solutions:


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We put all the blues on the diagonal, so none of them were touching, and then we put a red in one corner and a yellow in the other so they weren't touching and then we had to put the other colours where they are. Then we did the same with a red diagonal and a yellow diagonal:
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A great way of working out some possible results!

Then Danny told us how he worked out how many possible combinations there were:

First I worked it out starting with blue in the top left corner and red next to it in the top middle.
There are six different ways of starting (blue then red, blue then yellow, red then blue, red then yellow, yellow then red, yellow then blue) so there will be six times as many answers as I got starting with blue then red. This is because if I take one answer and swap all the blues for yellows and yellows for blues I get another answer, and there are six different ways to swap the colours.
The two square below the starting two can be yellow then blue, red then blue or red then yellow. That way I had three different ways the top left 2 x 2 squares could be coloured.
Then I worked out how many different ways you can colour in the top right and middle right squares.
There are three ways for each of the different starting 2 x 2 squares. For instance for the starting 2 x 2 with yellow then blue on the second row, you can have:
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Then I worked out how many ways the bottom left and bottom middle squares can be coloured. There are three ways for each of the different 2 x 3 top blocks, but some of them use too many blues or reds.For instance:

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uses too many blues. There are ten different ways of filling in all but the bottom right square, and then this one must be whatever square you've only used twice before. So there are ten different ways starting with blue then red, so there must be sixty different ways altogether.

Thank you, Danny!