## A British pensioner solves a long, open-ended mathematical puzzle

Mathematicians have a tendency to cover infinitely large floors with tiles that are supposed to have very special properties. They've been searching for the “Einstein Tile” for decades, but it has nothing to do with the famous physicist.

MMathematicians sometimes ask themselves strange questions, and the answers to them present great challenges. For example, there are the so-called Millennium Problems, which is a list of unsolved mathematical problems published in the year 2000. Clay Mathematics Institute (CMI) in Cambridge, Massachusetts – With a promise: Anyone who can solve one of these problems will receive a million-dollar prize.

So far only one millennium problem has been solved. In 2002, Russian mathematician Grigory Yakovlevich Perelman provided proof of its validity Poincaré conjecture Which has great sporting importance. Perelman became more famous than anyone expected based on his tremendous athletic accomplishments alone.

What caused a stir at the time was that this person, who had devoted his entire life to mathematics, actually refused to accept the prize money. For many people, this was completely incomprehensible, and some said he could have at least donated the money to a good cause if he didn't want it for himself.

The mathematical question of whether there was an “Einstein Tile” was not about big money. Mathematicians thought about this problem for almost 60 years until it was finally solved in 2022. Contrary to what you might think when you hear the term “Einstein's slab,” the mathematics in question has absolutely nothing to do with the famous physicist Albert Einstein.

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The question here is: Is there a specially shaped tile with straight edges that can be used to tile a floor of any size – mathematicians like to call it infinitely large – without leaving any gaps and without the resulting pattern repeating itself at some point, i.e. whether there is a period? So it comes down to whether you can handle the “single stone” – the “Einstein Tile” – to cover the floor as required.

That this is a very difficult task can be deduced from the fact that it took many years for the “Einstein Tile” to actually be found. Meanwhile, quite a few mathematicians believed that such tiles could not exist. But again, proof would be needed as to why the Einstein Tile does not exist. However, we now know that there can be no such evidence.

### My hat has 13 corners

It may have taken a layman's new perspective to discover Einstein's Tile. It was retired printing system technician David Smith, who lives in Great Britain, who achieved this mathematical feat. He turned to one of the scientists University of Waterloowho then, after careful examination, informed the scientific community of the discovery of the “Einstein Tile” in 2023.

What does the “Einstein Tile” look like? It is a 13-corner somewhat reminiscent of a crooked hat. That's why some people also refer to them as “hat tiles.”

It is difficult for anyone to have a tiled bathroom, but mathematicians are thrilled by such discoveries. Technically you are talking about an “aperiodic mono”. Once an example of such a tile was found, professional mathematicians were quickly able to show that there were an infinite number of variations of “Einstein tiles.”

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Its shape transitions are smooth, sometimes looking more like a hat, sometimes more like a turtle or a dragon. the National Museum of Mathematics New York even launched the “Einstein Mad Hat Awards” competition, which focused on creative interpretations of Einstein's tiles. There were 245 participants from 32 countries.

Even a British Nobel Prize winner Sir Roger Penrose I was looking for a tile that could be used to cover any large area without gaps and without repeating the pattern. He did not encounter the Einstein Tile, but at least in 1973 he discovered the Zweisstein Tile. With a set of just two different pieces, the desired design could be achieved, which is now referred to as “Penrose wrapping”.

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