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 $n$)<br/>
I conjecture that there are only $4$ types of such polynomino:<br/>
-The first trivial type is rectangle with integer side length.<br/>
-The second type is created by $3$ rectangles with integer side as follow:<br/>
[![enter image description here][1]][1]<br/>
The last two type is polyomino similar to one of two polyominoes below:<br/>
[![enter image description here][2]][2]<br/>
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 $n$?<br/>
Here some link about this problem:<br/>
-http://www.recmath.org/PolyPages/PolyPages/index.htm?RepO.htm<br/>
-https://www.wcupa.edu/sciences-mathematics/mathematics/vNitica/webRepTiles.aspx<br/>
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.


  [1]: https://i.sstatic.net/exsdW.png
  [2]: https://i.sstatic.net/3gll8.png