The problem of 36 officers
In fact, this was put forward by the great mathematician Euler. The main content is to select six officers with different ranks from six different legions to form a square with six rows and six columns. So these six officers from all walks of life happen to come from different legions and have different ranks. How should this square be arranged? If (1, 1) is used to represent the officer with the first rank from the first legion, (1, 2) is used to represent the officer with the second rank from the first legion, and (6,6) is used to represent the officer with the sixth rank from the sixth legion, then Euler's problem is how to arrange these 36 pairs into a square, so that. Historically, this problem has been called the 36-official problem.
solve
After the question of 36 officers at that time was raised, it remained unresolved for a long time. It was not until the beginning of the 20th century that it proved impossible to arrange such a parade. Although it is easy to generalize the number of regiments and ranks in the 36-official problem to the general N cases, the corresponding square satisfying the conditions is called N-order Euler square. Euler once guessed that for any non-negative integer t, the Euler square of order n=4t+2 does not exist. When t= 1, this is a problem for 36 officers, while when t=2, n= 10, mathematicians have constructed Euler squares of order 10, which shows that Euler conjecture is wrong. But by 1960, mathematicians had completely solved this problem and proved the existence of Euler squares of order n=4t+2(t≥2).
App application
This kind of square matrix is called orthogonal latin squares in modern combinatorial mathematics, which is widely used in industrial and agricultural production and scientific experiments. It has been proved that all orthogonal Latin orders 3, 4, 5, 7, 8, ... can be produced except the second and sixth orders.