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Veronica Phan
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Polyomino that can tile itself

Find all polyomino $P$ such that we can tile $nP$ with $n^2$ copies of $P$ for all $n\in \mathbb{N}$. ($nP$ is a polynomino similar to $P$ with scale factor $n$)
I conjecture that there are only $4$ types of such polynomino:
-The first trivial type is rectangle with integer side length.
-The second type is created by $3$ rectangles with integer side as follow:
enter image description here
The last two type is polyomino similar to one of two polyominoes below:
enter image description here
Is there any polyomino which not in the $4$ types above and satisfies that property? What if we replace polyomino by polyiamond, polyabolo, polycube,...? And as Timothy suggest, what if we allow all but finitely many $n$?
Here some link relate to this problem, but none has any attempt to solve it:
-More rep-tile polyominoes
-The same question but with polygonal
I think for too complex polyomino, combine copies of it would make either more complex polyomino or too simple polyomino, so it can't tile itself. So I guess we need some invariants which measure the complexity of polyomino. And $n=2$ may be the most important case.

Veronica Phan
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