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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 !

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    The tiles you are talking about are called *polyominoes*. There is a lot of literature about them. I don't know whether the specific problem you ask about has been studied.2012-09-27
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    Oh yes, I forgot the name, thanks! At the moment I don't know many things on them. Do you know if there is a place where all solved problems and references are listed?2012-09-27
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    Are those $n$s in $S_n$ and the number os squares the same? If not, try changing one of them, this can be misleading.2012-09-27

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