Math Trivia: Interlocked Cubes

Here is a math fact that may surprise you:

There exists a configuration in three dimensional space of a finite number of unit cubes such that no cube can be removed without moving the others. That is to say it is not possible to hold all but one of the cubes fixed in space and move the last cube arbitrarily far away from the others.

This should surprise you. Your intuition is probably saying “Why don’t you just take the highest cube and move it straight up?” That intuition would be correct if we were talking about spheres, but not with cubes. A cube that is on average below another cube can still be partially above that cube.

The configuration is actually not all that difficult to imagine. First notice that if you balance a cube on one of its corners and take a horizontal cross section right in the middle, that cross section is a regular hexagon. You can take a tiling of the infinite plane by hexagons, and extend that tiling to cubes. If we tile the xy plane with these hexagons, and extend it to cubes, all of those cubes will have the same z coordinate. Each cube will be touching 6 other cubes, and even though they have have the same height, 3 of those are positioned “above” the cube to stop it from going ¬†up, while 3 of them will be positioned “below” the cube to stop it from going down. If this is hard for you to imagine, or you want to see a proof, you can see it in this paper. (Look at figure 6)

Now that you believe that you can tile a plane, instead tile a very large square with cubes, so that no cube can be moved up or down. Make all the cubes very slightly smaller, so that you have some freedom to bend this large square. Wrap it around to make a large cylinder, then wrap it around in the other direction to make a large torus. When you wrap it around, you will glue edges of the square to other edges of the square, by interlocking the cubes on the boundaries.

The final configuration is a large torus, which locally looks like a plane tiled by the hexagonal cross sections of the cubes, but with the cubes slightly shrunken.