Simple Application of Hall's Theorem

Question

On opposite sides of a piece of paper there are drawn maps in which the same number of countries appear, all of equal area. Prove that one can pierce the sheet of paper with a pin a number of times in such a way that each country (on either side ) is pierced exactly once.

Answer 1

Let the countries on one side be the men of Hall's Marriage Theorem, let the countries on the other side be the women. Call a man and a woman compatible if there a place to put a pin so it goes through both of them. Piercing each country exactly once is then the same as marrying off compatible couples, so it can be done if Hall's condition is satisfied.

Now take any set of men. They cover a certain area. That area must overlap at least the same number of women, since each of the men and women has the same area. So Hall's condition is satisfied.