12 Apr 2013

Prize Maths Quiz: Minimal Tilings (PMQ14)


Today’s PMQ comes in two parts. Firstly, take a square with sides 7 units in length. You are required to tile this with smaller squares, each with whole number sides, in such a way as to use the minimum number of smaller squares. One extra condition is that the distinct tiles used must have no common factors, or put another way, the gcd of any two different tiles must be equal to 1.

The diagram shows one possible tiling for a square with sides of 6 units. At one extreme, we could just use unit squares, in which case we would need 36 tiles. At the other extreme, we could use one tile with sides 5 units and 11 unit tiles, making a total of 12. Neither of these will give us the minimum number of tiles; they are just presented as examples of how to approach this problem. However, notice that the tiling in the diagram satisfies the second condition. There are three distinct tiles used - of sides 1, 2 and 3 - and their pairwise gcd equals 1. There are solutions that use tiles of sides 2 and 4 but this would give a pairwise gcd of 2.

The second step in the question is to find all the different arrangements of the minimum number of tiles. These must be unique layouts, so that tilings with rotational symmetry count as just one arrangement. If we were actually tiling a floor, each arrangement could be rotated four times to give the appearance of different layouts, but it is still just a single unique design.

To recap the questions, you are given a square with sides of 7 units. You must find the minimum number of smaller squares with integer sides that can completely cover this larger square with no overlaps and no gaps. The sides of the distinct squares used can have no pairwise common factor other than 1. Then, using these minimum number of squares, how many unique arrangements are there?


[edited for clarity on 13 April]



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