IMO 1967 LL ITA27

Origin: ITA

Problem

Which regular polygons can be obtained (and how) by cutting a cube with a plane?

Solution

Regular polygons with 3, 4, and 6 sides can be obtained by cutting a cube with a plane, as shown in the figure. A polygon with more than 6 sides cannot be obtained in such a way, for a cube has 6 faces. Also, if a pentagon is obtained by cutting a cube with a plane, then its sides lying on opposite faces are parallel; hence it cannot be regular.