# Euler’s 243-year-old ‘impossible’ puzzle gets quantum solution

Quantum Latin squares were quickly adopted by a group of theoretical physicists and mathematicians interested in their unusual properties.Last year, French mathematical physicists Ion Nechita Jordi Pillet created a quantum version of Sudoku –sudo. In SudoQ, instead of using the integers 0 to 9, rows, columns and sub-squares have nine vertical vectors each.

These advances lead to Adam BurchartA postdoctoral researcher at Jagiellonian University in Poland and his colleagues revisited Euler’s old mystery about the 36 officers. What if, they wondered, Euler’s officers were quantum?

In the classic version of the problem, each entry is an officer with a clear rank and regiment. It is helpful to think of the 36 officers as colorful chess pieces, their rank can be king, queen, rook, bishop, knight or chess pieces, and their legions are represented by red, orange, yellow, green, blue or purple. But in the quantum version, officers are formed by the superposition of rank and regiment. For example, an officer can be a superposition of a red king and an orange queen.

Crucially, the quantum states that make up these officers have a special relationship called entanglement, which involves correlations between different entities. For example, if a red king is entangled with an orange queen, then even if both the king and queen are in a superposition of multiple clumps, observing that the king is red, you’ll immediately know that the queen is orange. It is because of the peculiarity of entanglement that the officers of each line can be vertical.

The theory seems to work, but to prove it, the authors had to construct a 6-by-6 array full of quantum officers. The sheer number of possible configurations and entanglements meant that they had to rely on the help of computers. The researchers inserted a classical approximate solution (an arrangement of 36 classical officers with only a few repetitions of ranks and regiments in a row or column) and applied an algorithm that adjusted the arrangement to a true quantum solution. The algorithm works a bit like solving a Rubik’s cube with brute force, you fix the first row, then the first column, the second column, and so on. As they repeat the algorithm over and over, the puzzle array gets closer and closer to the real solution. Eventually, researchers got to the point where they could see patterns and manually fill in the few remaining entries.

In a sense, Euler was proven wrong—though in the 18th century he couldn’t have known the possibility of quantum officials.

“It’s good that they closed the book on this issue,” Nechita said. “It’s a really beautiful result and I love the way they got it.”

According to co-author Sohail Rather, a physicist at the Indian Institute of Technology, Madras, Chennai, a surprising feature of their solution is that officer ranks are only associated with adjacent ranks (kings and queens, white rooks and bishops, knights and chess pieces) and regiments are entangled. with adjacent legions. Another surprise was the coefficients that appeared in the quantum Latin square. These coefficients are essentially numbers that tell you how much weight to give different terms in the stack. Strangely, the ratio of the coefficients used by this algorithm is Φ, which is 1.618…the famous golden ratio.