Three Way Mix Up


We received several different arrangements of tiles - thank you to those who sent in your suggestions. You had to remember that you didn't necessarily need one tile of each colour in every row and column - it was just that tiles of the same colour weren't allowed to touch. Not many of you looked for more than one other arrangement but Kesavan from Latymer All Saints C of E Primary sent in these solutions:





Louise from Farnborough Hill sent these arrangements:




Trish and James sent 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!

Finding all the possible arrangements of tiles is quite a challenge and some of you began to have a go at this. 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 squares 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 two by two 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 two by two squares.

For instance for the starting two by two squares with yellow then blue on the second row, here are the first and second rows of the whole arrangement:
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Then I worked out how many ways the bottom squares can be coloured and I think there are a total of three ways which work with this arrangement of the top left two by two square.

Then I looked at the arrangements of still having blue and red in the top left and top middle, but this time with red and blue underneath them, but none of these worked.

Finally, I looked at the arrangements of blue and red in the top left and top middle with red and yellow underneath them and I found another three ways.

So, I think there are six different ways of starting with blue then red, so there must be thirty six different ways altogether.

Thank you, Danny!