Call $S_n$ the square of area $n^2$. See it as a collection of $n^2$ unit squares. In the following, what I call tile is a collection of unit squares that are glued together.
If $n$ is not prime, say $p \times q$, it is possible to tile $S_n$ with $n$ tiles that are rectangles whose sides are $1$ and $n$. It is also possible to tile $S_n$ with $n$ rectangles whose dimensions are $p$ and $q$.
So, when $n$ is not prime, there is not a unique way to tile $S_n$ with exactly $n$ tiles of the same shape.
For $n=2$, $3$ or $5$, easy computations show that the rectangle of dimensions $1$ and $n$ is the unique shape that tiles $S_n$ with $n$ elements. What about other primes?
I don't know if this question is well-known and/or has been studied. I have looked at several chapters of Martin Gardner's books but I did not found this one.
Thanks by advance for your comments !