Hole in a cube
Take a cube, and hold it in such a way that one of its corners points directly towards you and the opposite corner is directly behind it. The cube will look like a regular hexagon.
One can make a square perforation through this hexagon through which one can pass another cube. The largest square that fits inside the hexagon has sides equal to √6 – √2 = 1.035... times the side of the cube. That means that we can make a hole on a cube and pass through it a larger cube! This surprising fact is mentioned by Martin Gardner in More Mathematical Puzzles and Diversions, Chapter 1: The Five Platonic Solids (Penguin Books, 1966).
In order for the rest of the cube to remain one piece, the hole has to be smaller than √6 – √2.
Here's a blueprint of a paper model of the cube with a hole with side exactly 1. Draw it on a sheet of paper, cut it out and assemble it with Scotch tape.
Click on the image above to get a GIF that just fits in a US Letter page, for printing out.
The finished object should look like this:
(This model was done with Microspot 3D World™ 3.1 Demo. Very cool little program ;-)
If you make two of these, each one will be able to pass through the other, at least in theory.