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Peculiar Perimeter

Problem

From the diagram below we can see that the number of tiles on the perimeter, 16, exceeds the number of tiles on the inside, 8.


How many rectangles exist for which the number of tiles on the perimeter are equal to the number of tiles on the inside?


Solution

If the rectangle measures m times n, the number of tiles on the perimeter is 2m + 2n minus 4.

For the number of tiles inside to equal the number of tiles on the perimeter, they must both be equal to half the area.

Therefore
mn
2
= 2m + 2n minus 4, which gives mn minus 4m minus 4n + 8 = 0

By writing this as mn minus 4m minus 4n + 16 = 8, we get (m minus 4)(n minus 4) = 8.

As m and n are both integer we obtain the factor pairs 1 times 8 and 2 times 4. Hence the only two solutions are rectangles measuring 5 times 12 and 6 times 8.

Problem ID: 89 (Nov 2002)     Difficulty: 3 Star

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