You are at a party, serving a square cake (with a thin layer of frosting on the top and edges) to 7 people. They each want an equal amount of the cake, and to make things worse, each want an equal amount of the frosting.

You try to figure out how do do this and eventually give up and simply try to cut the cake so each person gets the same amount of the frosting, ensuring that each cut starts at the center.

When you pass out the cake, you expect complaints, but when there are none, you realize that as luck would have it, you managed to cut into pieces of equal area.

How did this happen, and would it happen with more or fewer people?

Just divide the perimeter in N equal parts, starting wherever you please, and always cut to the center. It's easy to see that each person will receive a part that can be divided in triangles, all with the same height, and bases that sum to (perimeter/N), thus having the same area.

This would also work for any regular polygon cake: equilateral triangles, pentagons, hexagons, and so on.

Posted by Old Original Oskar! on 2006-10-23 09:28:01