This question was communicated to me by Evgeniy Romanov.

Consider a connected polyomino $P$ that can be completely tiled in two different ways: with disjoint $2 \times 2$ square tetraminoes, and with disjoint S-shaped tetraminoes (that we allow to reflect and rotate arbitrarily). Is it true that $P$ can not be simply connected, that is, $P$ must contain a "hole" in it?

The smallest non-trivial example of $P$ is a wedge of four $2 \times 2$ squares:

```
..AA.
BBAA.
BB.CC
.DDCC
.DD..
```

One can see that $P$ can be tiled by S-shapes as follows:

```
..AA.
BAAC.
BB.CC
.BDDC
.DD..
```

A brute-force shows that there is no counter-example fitting into an $9 \times 9$ square.