## Rugs

From the 1998 University of Waterloo Fermat Contest:

Three rugs have a combined area of 200 sq m. By overlapping rugs to cover a floor of area 140 sq m, the area which is covered by exactly two layers of rug is 24 sq m. What area of floor is covered by three layers of rug?

Here it is helpful to write down something wrong and then correct it:

INCORRECT: 140 = area of rug A + area of rug B + area of rug C.

This is wrong because on the right hand side, the area covered by exactly two layers of rug (which is 24) has been counted twice, and the area covered by three layers of rug (call this x) has been counted three times. So we can correct it like so:

CORRECTION: 140 = area of rug A + area of rug B + area of rug C – area covered by exactly two rugs – 2(area covered by exactly three rugs).

That is,

140 = 200 – 24 – 2x.

So x = 18, and the total area covered by three layers is 18 sq m.

So simple when you see it written down, yet really quite tricky to formulate from scratch. A great problem.