Reporting by XinZhiyuan
EDIT: LRS
243 years ago, Euler left the famous "Thirty-Six Officers" puzzle, which has not been perfectly proven. Recently quantum physicists have said they have solved this unsolved mystery, but only require the officer to be in a quantum state!
In 1779, the famous Swiss mathematician Leonhard Leonhard Euler once asked the question: how should 6 officers of 6 different ranks from different 6 army regiments be selected from each of the 6 army regiments in a platoon of 6 rows and 6 columns, so that the 6 officers in each row happen to be from different corps and have different ranks.
If (1,1) is used to represent officers from the first legion with the first rank, (1,2) to mean officers from the first corps with the second rank, and (6,6) to mean officers from the sixth corps with the sixth rank, then Euler's problem is how to arrange the 36 pairs of numbers into a phalanx so that the numbers per row and column are made up of exactly 1, 2, 3, 4, 5, and 6, both in terms of the first number and the second number.
This question, historically known as the "Thirty-Six Officers' Question," was not answered for a long time.
In fact, when the rank and number of legions is 5 or 7, this problem becomes very easy to solve, but the solution of thirty-six officers is not found.
So in 1782, Euler said that after painstaking efforts, although it was impossible to give rigorous proof, it was necessary to admit that such a arrangement (placing 36 officers in this form in the square of 6×6) was impossible.
At the time, Euler proved that for this puzzle, any number of legions and ranks that do not exist in the form of 4k+2 exists. He stated that the method of proof he had adopted was not suitable for proof of such a form of figure.
It wasn't until more than a century later, in 1901, that the French mathematician Gaston Gaston Tarry proved that there was really no way to arrange Euler's 36 officers in a 6×6 square without repeating, and he wrote all possible arrangements of the 6x6 square, proving that the 36 officer problem was impossible.
Fast forward to 1960, and mathematicians, with the help of the computer, proved that this puzzle had a solution to any number of legions and ranks greater than 2, except for 6.
For example, the following figure shows a 5×5 square that can be filled with 5 different levels and 5 different colors of pieces, and there are no duplicate levels or colors in the same row or column.
By 1960, mathematicians had used computers to prove that there was a solution to any number of ranks and legions, except for 6 ranks and 6 corps.
Indecisive, quantum mechanics
The Thirty-Six Officer Problem looks very similar to the Sudoku game, but mathematically there is another classification of these two types of puzzles.
Sudoku is a "Latin square matrix", that is, a square matrix of symbols (numbers and letters), in which each symbol appears only once in each row and column. If you combine two Latin squares with the same size but different symbols, you get a Greek Latin square, also known as euler squares, the main feature of which is that they contain pairs of symbols.
So if the solution to the thirty-six officer problem exists, it must be a 6x6 Greek-Latin phalanx, with pairwise attributes of rank and legion.
Recently, in a paper in the Physical Review Letters, quantum physicists from the Indian Institute of Technology (Madras Polytechnic Campus), Jagiellonian University and other institutions proved that using the idea of quantum mechanics, it is possible to arrange these 36 "quantum version of the officers" into a grid in a way that meets Euler's standards.
Address of the paper: https://arxiv.org/abs/2104.05122
In the classic version, each grid in the phalanx represents an officer with a definite rank and corps.
But in the quantum mechanical version, particles like electrons can be in a "superposition" of multiple possible states, and the officers in the quantum version are made up of the superposition of its ranks and legions. For example, an officer can be a red king and an orange queen superposition.
Crucially, the quantum states that make up these officers are entangled. For example, if a red king is entangled with an orange queen, then even if the king and queen are in the superposition of multiple legions at the same time, it can be determined that the king is red and the queen is orange. Because of this particular entanglement property, officers on each row or column can be perpendicular to each other.
To prove this theory, the researchers needed to construct a phalanx of 6×6 filled by these quantum officers. Since there are a large number of possible combinations and entanglements, they must rely on quantum computers.
In the phalanx, the researchers first enter an approximation of the classic version of the puzzle. In this approximation solution, the arrangement of the 36 classic officers in a row or column has only a small number of ranks and corps repetitions.
They then applied an algorithm to the solution that would adjust this arrangement to a true quantum solution. This algorithm works a bit like solving a Rubik's Cube with brute force - fix the first row first, then fix the first column, fix the second column... As they repeat this algorithm over and over again, they can get closer and closer to the real solution.
Using this algorithm, they ended up with a real solution to the mystery of 36 officers. In a sense, it proves that Euler's judgment of the 36 officers' puzzle is "wrong".
But what is certain is that Euler in the 18th century could not have imagined that officers could still be "quantized".
It is worth mentioning that the new solution has a feature, that is, the rank of officers is only entangled with adjacent ranks, such as king and queen, chariot and elephant, horse and soldier, and the legion is only entangled with the adjacent corps. And the coefficient ratio in the quantum Latin square is also 1.618, the famous golden ratio.
Of course, for 243 years mathematicians have not only been after a puzzle answer, the solution is also known as the Absolute Maximum Entangled State (AME), which solves the problem of arranging quantum objects and is important in many applications, including quantum error correction, such as providing a way to store redundant information in quantum computers so that even if the data is damaged, the information can be preserved.
In AME, there is also a strong correlation between the measured values of quantum objects.
In the case of a coin toss, if two people (A, B) toss a tangled coin, and A tossed the coin and got the front, he would know that Bob was the opposite, and vice versa. Two coins can be entangled to the maximum, three can be, but four can't: if two people join the coin toss together, A never knows what B got.
The paper's research proves that if you have a set of four quantum-entangled dice instead of ordinary coins, they can be entangled to the greatest extent possible. The arrangement solution for the six-sided dice is equivalent to a 6×6 quantum Latin square.
Because of the presence of a golden ratio of 1.618 in this solution, the researchers also refer to it as "golden AME."
Resources:
https://www.quantamagazine.org/eulers-243-year-old-impossible-puzzle-gets-a-quantum-solution-20220110/
https://m.huxiu.com/article/490056.html