WWII Planes

During World War II, the mathematician Abraham Wald was asked to help with determining which parts of the allied forces’ planes must be armored better. After examining the surviving American planes, he noticed that there were many holes in the fuselage, and very few in the engines. After careful thinking, he suggested that the armor on the engines must be improved. Why?

Abraham Wald realized that the holes should have been distributed more evenly across the planes. Therefore the planes which had more holes in the fuselage survived, while the planes which had more holes in the engines got destroyed.

Cut the Paper

You have a sheet of paper and a rectangular piece cut from it, as shown in the picture. What is the easiest way to cut the paper into two pieces with equal areas?

Make a cut through the centers of the small and the big rectangle. The cut will split both the area of the big and the small (absent) rectangle by half, and therefore will do the same to their difference – the given sheet of paper.

Square Cake

There is a square cake at a birthday party attended by a dozen people. How can the cake be cut into twelve pieces, so that every person gets the same amount of cake, and also the same amount of frosting?

Remark: The decoration of the cake is put aside, nobody eats it.

Divide the boundary of the cake into twelve equal parts, then simply make cuts passing through the separation points and the center. This way all tops and bottoms of the formed pieces will have equal areas, and also all their sides will have equal areas. Since all pieces have the same height, their volumes will be equal as well.