Question

29. My grandson makes wall hangings by stitching

together 16 square patches of fabric into a 4 x 4
grid. I asked him to use patches of red, blue,
green and yellow, but to ensure that no patch
touches another of the same colour, not even di-
agonally.
The picture shows an attempt which fails only
because two yellow patches touch diagonally.
In how many different ways can my grandson
choose to arrange the coloured patches correctly?
​

Answer 1

9514 1404 393

Answer:

168

Explanation:

Below are shown the 7 different kinds of arrangements that can be made. Each of these can be permuted into 24 different arrangements by assigning colors in a different order than the one used here:

{1, 2, 3, 4} ⇒ {green, blue, red, yellow}

Then the total number of possible correct colorings is 7·24 = 168.

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These colorings were found using a computer program that identified all 168 viable colorings, then eliminated permutations. It started by identifying ways to color 2 rows, then adding rows based on that.

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You'll notice that the last on the attached list most closely matches the one in the problem statement.