Is there a single aperiodic tile?
If you had infinitely many square tiles of the same size, you could tile an infinitely large flat floor with these tiles, each fitting tightly with those around it, and the result would be periodic, which means that no matter where you looked at the floor, the tiling pattern would look the same. More formally, the tiling pattern is periodic because if you shift the whole pattern to the left or right, or up or down, by one tile width, it will look exactly like it looked before you shifted it. If you started with regular hexagonal tiles you would also be able to tile the floor with a periodic tiling pattern. If you started with infinitely many regular hexagonal tiles and infinitely many equilateral triangle tiles, with the same side lengths, you could also tile the floor with a periodic pattern.
The first amazing fact is that there are some sets of tiles with which you can tile your infinite floor, but such that no matter what tiling pattern you make, that pattern will never be periodic, it will never repeat itself. There will be no way to shift your pattern in some direction and have the pattern look the same as it did before you shifted it. This is called an aperiodic set of tiles. The famous "Penrose tilings" are examples of tilings created with such a set: there are two different tile shapes, neither of which can tile a floor by itself, such that if you use both of them you can tile the infinite floor, but not periodically. The question is: Is there a single tile shape that tiles the infinite floor, but that cannot tile the infinite floor periodically? Such a tile is called a single aperiodic tile, or an aperiodic monotile.
Click on the image to the right to see examples of both periodic and aperiodic tilings.
(Thanks to Vinay Kathotia for explaining this problem to us.)