# Holes in tileable double polynomino

This question was communicated to me by Evgeniy Romanov.

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

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

``````..AA.
BAAA.
BB.CC
.DDCC
.DD ..
``````

We can see that $$P$$ can be tiled by S-forms as follows:

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

A brute force shows that there is no counterexample corresponding to a $$9 times 9$$ square.