Math Puzzle: Falling Through

"Some say the reason most manhole covers are round is that a circle cannot fall through a smaller circular hole. Which of these other two-dimensional shapes cannot fall through a hole that is the same shape but slightly smaller? Shapes 1, 2 and 3 can all fall through their own holes. Shape 4 cannot. Challenge problem: Can you find another shape that cannot fall through a slightly smaller hole of the same shape?"

"The circle is one of an infinite family of shapes that have a constant diameterthat is, no matter what angle a line crosses it at, the widest point of the shape will always be the same. The Reuleaux triangle is another example and thus another shape that cannot fall through a slightly smaller hole of the same shape."

Most manhole covers are round because a circle cannot fall through a smaller circular hole. Some two-dimensional shapes can pass through a hole cut in a slightly smaller copy of themselves; shapes 1, 2, and 3 are such examples while shape 4 cannot. Shapes with constant width keep the same maximum distance across at every orientation, so they cannot be rotated to pass through a slightly smaller congruent hole. The Reuleaux triangle is another constant-width shape that cannot fall through a smaller copy. Shapes that can pass through copies of themselves are called Rupert; the noperthedron is the first proven non-Rupert polyhedron of its kind.

Read at www.scientificamerican.com