Euler's Officers
How many different ways can you arrange the officers in a square?
Age
14 to 16
Challenge level
Problem
How many different solutions can you find to this problem?
Arrange 25 officers, each having one of five different ranks $a$, $b$, $c$, $d$ and $e$, and belonging to one of five different regiments $p$, $q$, $r$, $s$ and $t$, in a square formation 5 by 5, so that each row and each file contains just one officer of each rank and just one from each regiment.
Getting Started
See Teddy Town
You can construct orthogonal Latin squares $S^{i,j}$ and $T^{i,j}$ of prime order $m$ where the $S^{i,j} = si + j \pmod m$ and $T^{i,j} = ti + j \pmod m$ and $s$ not equal to $t$.
Student Solutions
This has proved to be a Tough Nut. Reading the article on Latin Squares published in September 2002 should help you to solve this.
Taking $s=1$, $2$, $3$ or $4$ you can construct $4$ different Latin squares $S^{i,j}$ of order $5$ where $S^{i,j}=si+j \pmod 5$.
Now suppose the numbers are used to denote the five ranks and consider how many different arrangements there will be if no two officers of the same rank or of the same regiment appear in the same row or in the same column.