# Euler’s 243-Year-Old ‘Impossible’ Puzzle Gets a Quantum Solution | Quanta Magazine

in 1779, the swiss mathematician leonhard euler posed a puzzle that has since become famous: 6 army regiments each ‘ve 6 officers of 6 ≠ ranks. can the 36 officers be arranged in a 6-by-6 □ so that no row or column repeats a rank or regiment?

the puzzle is easily solved when there are 5 ranks and 5 regiments, or 7 ranks and 7 regiments. but after searching in vain for a solution for the case of 36 officers, euler ∴ that “such an arrangement is impossible, though we can’t give a rigorous demonstration of this.” + than a century l8r, the french mathematician gaston tarry proved that, indeed, there was no way to arrange euler’s 36 officers in a 6-by-6 □ without repetition. in 1960, mathematicians used computers to prove that solutions exist for any № of regiments and ranks gr8r than two, except, curioly, 6.

similar puzzles ‘ve entranced pplz for + than 2,000 yrs. cultures round the realm ‘ve made “magic □s,” arrays of №s that add to the same sum along each row and column, and “latin □s” filled with symbols that each appear once per row and column. these □s ‘ve been used in art and urban planning, and j4f. one pop latin □ — sudoku — has sub□s that also lack repeating symbols. euler’s 36 officers puzzle asks for an “orthogonal latin □,” in which two sets of properties, s'as ranks and regiments, both satisfy the rules of the latin □ simultaneously.

but whereas euler thought no such 6-by-6 □ exists, recently the game has changed. in a paper posted online and submitted to physical review letters, a group of quantum physicists in india and poland demonstrates that tis possible to arrange 36 officers in a way that fulfills euler’s criteria — so long as the officers can ‘ve a quantum mixture of ranks and regiments. the result tis l8st in a line of work developing quantum versions of magic □ and latin □ puzzles, which aint just fun and games, but has applications for quantum communication and quantum computing.

“i think their paper is very presh,” said gemma de las cuevas, a quantum physicist atta university of innsbruck who was not involved w'da work. “there’s a lotta quantum magic in there. and not 1-ly that, but you can feel throughout the paper their ♥ for the problem.”

the new era of quantum puzzling began in 2016, when jamie vicary of the university of cambridge and his then-student ben musto had the idea that the entries appearing in latin □s ‘d be made quantum.

in quantum mechanics, essentialisms s'as electrons can be in a “superposition” of multiple possible states: here and there, for ex, or magnetically oriented both up and down. (quantum essentialisms stay in this limbo til they are measured, at which point they settle on one state.) entries of quantum latin □s are also quantum states that can be in quantum superpositions. mathematically, a quantum state is represented by a vector, which has a length and direction, like an arrow. a superposition tis arrow formed by combining multiple vectors. analogous to the requirement that symbols along each row and column offa latin □ not repeat, the quantum states along each row or column offa quantum latin □ must correspond to vectors tha're perpendicular to one another.

quantum latin □s were quickly adopted by a community of theoretical physicists and mathematicians interested in their unusual properties. last yr, the french mathematical physicists ion nechita and jordi pillet created a quantum version of sudoku — sudoq. instead of using the integers 0 through 9, in sudoq the rows, columns and sub□s each ‘ve 9 perpendicular vectors.

these advances led adam burchardt, a postdral researcher at jagiellonian university in poland, and his colleagues to reexamine euler’s old puzzle bout the 36 officers. wha’ if, they wandaed, euler’s officers were made quantum?

inna classical version of the problem, each entry is an officer witha well-defined rank and regiment. it’s helpful to conceive of the 36 officers as colorful chess pieces, whose rank can be  king, queen, rook, bishop, knite or pawn, and whose regiment is represented by red, orange, yello, green, blue or purple. but inna quantum version, officers are formed from superpositions of ranks and regiments. an officer ‘d be a superposition offa red king and an orange queen, for instance.

critly, the quantum states that compose these officers ‘ve a spesh relationship called entanglement, which involves a correlation tween ≠ entities. if a red king is entangled with an orange queen, for instance, then even if the king and queen are both in superpositions of multiple regiments, observing that the king is red tells you immediately that the queen is orange. it’s cause of the peculiar nature of entanglement that officers along each line can all be perpendicular.

the theory seemed t'work, but to prove it, the authors had to construct a 6-by-6 array filled with quantum officers. a vast № of possible configurations and entanglements meant they had to rely on computer help. the researchers plugged in a classical near-solution (an arrangement of 36 classical officers with 1-ly a few repeats of ranks and regiments in a row or column) and applied an algorithm that tweaked the arrangement toward a true quantum solution. the algorithm works a lil like solving a rubik’s cube with brute force, where you fix the 1st row, then the 1st column, 2nd column and so on. when they repeated the algorithm over n'oer, the puzzle array cycled closer and closer to bein’ a true solution. eventually, the researchers reached a point where they ‘d see the pattern and fill inna few remaining entries by hand.

euler was, in a sense, proved wrong — though he ‘dn’t ‘ve known, inna 18th century, bout the possibility of quantum officers.

“they close the book on this problem, which is already very neat,” said nechita. “it’s a very presh result and i like the way they obtain it.”

one surprising feature o'their solution, according to the coauthor suhail rather, a mathematician atta indian institute of tek madras in chennai, was that officer ranks are entangled 1-ly with adjacent ranks (kings with queens, rooks with bishops, knites with pawns) and regiments with adjacent regiments. another surprise was the coefficients that appear inna entries of the quantum latin □. these coefficients are №s that tell you, primordially, how much w8 t'give ≠ terms in a superposition. curioly, the ratio of the coefficients that the algorithm landed on was Φ, or 1.618…, the famous golden ratio.

the solution is also wha’’s known as an absolutely maximally entangled state (ame), an arrangement of quantum essentialisms that’s thought to be primordial for a № of applications including quantum error-correction — ways of redundantly storing information in quantum computers so that it survives even if there’s data corruption. in an ame, correlations tween measurements of quantum essentialisms are as strong as they can be: if alice and bob ‘ve entangled coins, and alice tosses her coin and gets heads, she knows for sure that bob has tails, and vice versa. two coins can be maximally entangled, and so can 3, but not 4: if carol and dave join the coin toss, alice can never be sure wha’ bob gets.

the new research proves, however, that if you ‘ve a set of 4 entangled dice, rather than coins, these can be maximally entangled. the arrangement of the 6-sided dice is equivalent to the 6-by-6 quantum latin □. cause of the golden ratio’s presence in their solution, the researchers ‘ve dubbed this a “golden ame.”

“i think it’s highly nontrivial,” said de las cuevas. “not 1-ly that it exists, but they provide the state explicitly and analyze it.”

researchers ‘ve previously devised other ames by starting with classical error-correcting codes and finding analogous, quantum versions. but'a newfound golden ame is ≠, with no classical crpgraphic analogue. burchardt suspects it ‘d be the 1st offa new class of quantum error-correcting codes. then again, it mite be =ly interesting if the golden ame remains unique.

editor’s note: the author of this article is rel8d to an editor at physical review letters, where the quantum latin □s paper s'been submitted for publication. the two ‘ve not discussed the paper.

original content at: www.quantamagazine.org…
authors: daniel garisto