One tile to rule them all

Here is a puzzle that stayed open for more than half a century, with a name that is half a joke. The Germans call it the einstein problem — not after the physicist, but from ein Stein, "one stone", one tile. The question is deceptively simple:

Is there a single tile shape you can lay down, copy after copy, that covers the whole plane with no gaps and no overlaps — but that can never settle into a repeating pattern?

Most shapes that tile do so periodically: shift the whole pattern by the right amount and it lands exactly on itself, like square floor tiles or a honeycomb. A tiling is aperiodic when no such shift exists — the pattern never repeats, no matter how far you slide it. The dream was a lone tile that is forced to be aperiodic: it simply cannot make a repeating pattern even if it tries. For fifty years nobody knew whether such a stone could exist. In 2023, an amateur found one.

The long march from thousands down to one

The story is really a story about a shrinking number. Mathematicians had long known you could force aperiodicity if you were allowed a whole set of different tiles — and the question became: how small can the set be?

• 1966 — 20426 tiles. Robert Berger proved the first aperiodic set exists, using Wang tiles (coloured squares whose edges must match). His set had a staggering 20426 tiles.
• Down to 13, then 11. Berger himself, then others, whittled the Wang-tile count down to 13, and later 11.
• Robinson — 6 tiles. In 1971 Raphael Robinson found a clean set of just 6 tiles that tile only aperiodically.
• 1974 — Penrose's 2 tiles. Roger Penrose famously got it down to 2 — the Penrose tiles (his "kite and dart", or two rhombi). This was the record for nearly fifty years.
• … and everyone wondered about 1. Two was so close to one. Could a single tile do it? Nobody could find one, and nobody could prove it was impossible.

Read that list downward and you can feel the pull toward the finish line: 20426 \to 13 \to 6 \to 2 \to \boxed{1}\,? The last step is the one that would not fall.

2023: an amateur finds "the hat"

In November 2022, David Smith — a retired print technician and hobbyist tinkering with shapes on his kitchen table — noticed that one particular 13-sided polygon seemed to tile the plane in a way that never quite repeated. He wrote to the mathematician Craig Kaplan. Together with Chaim Goodman-Strauss and software engineer Joseph Myers, the team proved in March 2023 that Smith's shape — nicknamed "the hat" — really is an aperiodic monotile. A fifty-year-old problem, cracked by a shape you could cut out of paper.

The hat is a single 13-sided polygon. It is a polykite: take one of the "kite" pieces you get by cutting a hexagon into six, and glue eight of them together. Step through the figure to see the tile, then its hidden kite skeleton, then the twist that comes next.

Worked example — the hat versus the spectre

The hat settled the problem… almost. To tile the plane, a hat pattern must use the hat tile together with its mirror image — the shape flipped over, like a left hand beside a right hand. For many people that was a perfect answer: it is still one shape, just used both ways up.

But purists objected. If you are only allowed to slide and rotate a tile — never flip it — then the hat and its mirror image are two different tiles, and the "one tile" claim wobbles. A tile that has a genuinely different mirror image is called chiral (from the Greek for "hand"). The open question sharpened to: is there a chiral aperiodic monotile, one that needs no reflections at all?

The same team answered within months. In May 2023 they announced "the spectre" — a single tile that tiles the plane aperiodically using only rotations and translations, never a reflection. (Their spectre uses gently curved edges so that a tile and its mirror can never accidentally fit together.) That is a true aperiodic monotile — one stone, no flipping. The ein Stein quest was finally, completely, over.