This deceptively simple question leads to some serious mathematical problems, including one of the biggest pieces of news in mathematics from the last ten years.

Imagine a floor or wall covering a very large area. As mathematicians work in unreal ideals rather than concrete physicalities, let us assume the area is infinite. How would we cover this infinite area with tiles all of the same shape, and what shape would they be?

The simplest answer coming to most people's minds would be the shape of a perfect square. We can align it nicely with our view of the world - north to south, east to west - and repeat it infinitely many times to the left and right to tile the entire plane in a perfect pattern. This tiling is periodic: if we picked up the whole infinite square grid of tiles and shifted it one square to the left, it would land perfectly on top of the same infinite square grid, with all the tiles aligning.

Other examples of periodic tilings are the triangular grid and the hexagonal grid. They look a little more complicated, to our rectangular human minds, but both of these tilings are also periodic: we can pick up the whole pattern and shift it so that it lands perfectly on top of itself. These three shapes - the square, the triangle, and the hexagon - are the only perfectly regular polygons which can each tile the entire plane by themselves, although there are many irregular polygons (with sides of different lengths or angles of different widths) which can also do so.

Some periodic tilings involve two or more different shapes. A commonly seen example is the grid of octagons, with squares in between them to fill the gaps. It's still periodic, because we can pick up the whole pattern, all those octagons and squares fitting together, and shift it to the left or right in order to land perfectly on top of itself again.

Much more difficult to visualise are aperiodic tilings. Aperiodic means "not periodic": is it possible to tile a floor in such a way that the whole area is covered without any gaps, but the pattern never repeats?

In the 1960s, American mathematician Robert Berger discovered the first aperiodic tiling, consisting of 20,426 different shapes. Obviously, this construction was neither elegant nor intuitive. Like many examples constructed by mathematicians, it was only made in order to prove someone else wrong! A mathematical question called the "domino problem" was proved to be undecidable by Berger's construction. But a set of 20,426 different tile shapes isn't going to help anyone to cover their kitchen floor. Nor is the set of 104 different tile shapes that Berger later constructed to improve his result.

Roger Penrose is a British mathematician, physicist, and philosopher, now 93 years old and a Nobel Laureate. One of his most famous discoveries in pure mathematics is the so-called Penrose tiling, a set of just two different tiles, each one being a diamond shape with just four sides, that form an aperiodic tiling of the plane. This tiling became known to geometry enthusiasts everywhere, and many mathematical institutions around the world have Penrose tiling visible somewhere, on a floor or wall, for those who enjoy such things.

The big question now remaining: is it possible to have just one single tile, one polygonal shape, which can only tile the plane aperiodically? Usually, in research-level mathematics, finding something like this is either very easy or impossible: if there are no simple examples, it's more likely that someone proves there are no examples than that someone finds a complicated example. But the problem of the "aperiodic monotile" remained unsolved by mathematicians for fifty years after Penrose's discovery of two tiles that together cover the plane aperiodically. It became known as the "einstein problem" - not named after the man Einstein, but after the German words "ein stein" meaning "one stone", or a single tile shape.

In 2022, retired British print technician David Smith discovered what he thought might be an aperiodic monotile. Smith is an amateur mathematician, without any professional position at a university or research institution, who simply loved mathematics for its own sake without making a career out of it. He contacted Craig Kaplan, a computer science professor in Canada, and later they were joined by Joseph Myers, a British software developer who is involved with the Cambridge mathematical scene without being employed as an academic mathematician, and Chaim Goodman-Strauss, a US mathematics professor, to publish a joint four-author paper which rigorously proved that Smith's discovery, a thirteen-sided polygon nicknamed "the hat" due to its shape, is indeed an aperiodic monotile and a solution to the einstein problem.

This is a rare example of a long-standing open mathematical question which has been resolved clearly within the last few years. When the news broke in 2023, it was exciting for the whole mathematical community. What other unsolved problems may be cracked next? Only time, and those who love mathematics, can tell.