Inclusions - Exclusions with areas

Question

Having problems solving this question:

Three carpets, each having an area of $3m^2$, are covering a $6m^2$ section of the floor. Show that some two carpets overlap on at least $1m^2$ of the floor

Any help tackling this problem would help.

Thank you

Answer 1

Hint: $6 = 3+3+3 - (a_{12}+a_{13}+a_{23}) + a_{123}$, hence $a_{12}+_{13}+a_{23} = a_{123}+3 \geq 3$. Therefore $\max(a_{12}, a_{13}, a_{23}) \geq 1$.

The estimate is sharp, as there is no restriction on the shape of the floor or of the carpets: consider a $2\times 3$ floor with the carpets covering the squares: $$((1,1), (1,2), (2,1))$$ $$((1,2), (1,3), (2,3))$$ $$((2,1), (2,2), (2,3))$$