# What can be tiled by T-tetrominoes?

The T-tetromino is a T-shaped figure made of four unit squares. An $m\times n$ rectangle can be tiled by T-tetrominoes if and only if both $m$ and $n$ are multiples of 4. This was proved in a 1965 paper by D.W.Walkup, and the proof was "hands on".

Some "algebraic" tricks like colouring or tiling groups can prove that $mn$ must be a multiple of 8, but they do not seem to rule out the cases like $99\times 200$ and $100\times 102$.

I wonder whether a better proof of D.W.Walkup's theorem is known today. By "better" I mean applicable to non-rectangular regions as well. For example, is there a way to determine what 6-gons (8-gons, ...) admit tiling by T-tetrominoes?

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The paper "Tiling rectangles with T-tetrominoes" by Korn and Pak seems to present some progress for non-rectangular simply-connected regions, mainly Theorem 11 in section 8. It doesn't seem like they have a complete answer though. citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.70.6847 – Alon Amit Jun 20 '10 at 21:57
See also Korn's thesis, "Geometric and algebraic properties of polyomino tilings". – Gjergji Zaimi Jun 21 '10 at 1:19

Now, motivated by the quest to find a better proof, I made a "local move connectivity" conjecture saying that every two T-tetromino tilings of a simply connected region are connected by a series of moves involving either two T-tetrominoes or four T-tetrominoes (forming a $4\times 4$ square). Usually, the "conceptual proof" comes from some kind of height function argument which also proves the local move connectivity. Now, Mike Korn in his thesis disproved this by a simple construction. One can ask if the Conway group approach in full generality can prove something like what you are asking. You need to compute $F_2/\langle tile~words\rangle$ (see Conway-Lagarias paper, "New horizons" or Korn's thesis). We did not do that, but I won't be very optimistic - it is a bit of a miracle when this approach works out.