Physicists have managed to crack an age-old puzzle without a solution by using principles from quantum mechanics.

Imagine six regiments, each of which has six officers with six different ranks, wrote the Swiss mathematician Leonhard Euler in 1779† Can you place it on a six-by-six chessboard in such a way that there are no two officers from the same regiment or rank in any row?

Who are equal here sudoku-skills If we want to let loose, we must disappoint. As Euler suspected and proved in 1901 by his French colleague Gaston Tarry: there is no solution to this riddle. And that’s crazy, because for every other number of regiments and officers, except two (try it), there is such a solution.

Physicists Suhail Ahmad Rather and Adam Burchardt have now, together with Indian and Polish colleagues, ‘cheated’ by using tricks from quantum mechanics on Euler’s puzzle. And then there is indeed a solution – which also immediately appeared to address an open issue from the world of quantum computers.


Important for the quantum solution of the ’36 officers riddle’ is that an officer can have multiple ranks and regiments at the same time. For example, he is a bit of a sergeant of regiment 1, and a bit of a captain of regiment 2. This is called a superposition, in this case of different ranks and regiments. Such a superposition then persists until you actually look at such an officer. Then he must choose a rank and regiment.

Compare it to a game of Stratego, where an opponent’s piece has an unknown rank, until you capture it with one of your pieces and you can turn it over. With the difference that the piece in this quantum-Stratego only gets its final rank and regiment when you hit it.

Tangled Pieces

In addition, there is entanglement between pieces. That is, they can have a quantum mechanical bond with each other, where what happens to one affects the other. A simple variant of such an entanglement is: two pieces are both a superposition of major and colonel. If one turns out to be a major when you look at him, the other must be a colonel.

Here too, an analogy can be made with Stratego. If your opponent has only two pieces left and you capture one, you immediately know the rank of the other. With quantum-Stratego, however, both pieces are actually ‘a little bit of one and a little bit of the other’, until you flip one of the two.

Not a normal piece

Back to Euler’s puzzle. The researchers started with a wrong solution of a riddle that came close to a good one. They then ran an algorithm to find a solution that allowed entangled pieces with multiple ranks and regiments. And sure enough: there turned out to be such a solution.

To do this, the can with quantum tricks had to be opened quite a bit. In the solution found, not a single piece is normal anymore. All of them are a superposition of at least two, and in most cases even four rank-and-regiment combinations.

Quantum Dice

Nice, of course, that such an unsolvable puzzle can still be solved in this way. It’s nice that there was – coincidentally – the same number of years between Euler’s original conjecture and Tarry’s mathematical proof, and between Tarry’s proof and the quantum solution of Rather, Burchardt and colleagues. But does it also help us?

Maybe. The quantum world has a similar problem. As a rule, quantum computers calculate with qubits: quantum bits that have a value of 0 or 1. But there are also ‘qutits’, which can have three values, and qubit variants with even more possible values. Here are especially important ‘quhexes’, which can have six values. Imagine them as a kind of quantum dice.

Absolutely maximally entangled

Now you can entangle those kinds of quantum dice in groups, just like before with the officers. One option is to ‘entangle them absolutely maximally’ (absolutely maximally entangled or AME). Then every set of dice you pick up is intertwined with what’s left on the table.

“If there are four dice, and Alice chooses and rolls two, she gets one of 36 equally likely outcomes,” the researchers explain in their scientific paper. “Bob then throws the other two. If the dice are absolutely maximally entangled, Alice can always infer Bob’s result from her own roll.”

‘Elegant solution’

At least… That would be the case if there were sets of four absolutely maximally entangled quantum dice. But just as you can’t get 36 officers on a six-by-six board without getting the same rank or regiment somewhere in a row or column, you can’t completely entwine such sets of four quantum dice.

Until now, because the approach that Rather and Burchardt used did yield such a set. “Its construction is a non-trivial optimization problem that the researchers solve in an elegant way in their article,” says physicist Barbara Terhalthat the theory group of the Delft quantum institute QuTech and was not involved in the investigation.

Correct mistakes

Now absolutely maximally entangled states play a role in all kinds of applications in the quantum world. When correcting errors in calculations of a quantum computer, for example, or when ‘teleporting’ particles: transferring the properties of one particle to another. And so now a new strategy has been devised to achieve such maximally entangled states.

physicist Hoi Kwong Lo of the University of Toronto, not involved in the study, calls the work, if it stands, “very important, with implications for correcting errors in quantum calculations” in a message from the American Physical Society

Terhal doesn’t want to go that far. “I wouldn’t say this is a big step in quantum error correction research. I don’t see any immediate applications. Rather, the aim of the research is to solve that puzzle. Is there such a crazy special condition?”