Primes of the D hexomino

4 × 6 (smallest rectangle)
5 × 12
complete

smallest rectangle: 4 × 6

Proposition. Any rectangle tiled by the D hexomino has one side divisible by 6 .
Proof. Consider the numbering

          (x, y) |---> {  1  if  x + [y/3]  is odd
                       { -1  if  x + [y/3]  is even,

    where  [ ]  is the greatest integer function.
  

However it is placed, a D hexomino covers a total of 0. It is easy to check that a (6m + 2) × (6n + 3) rectangle can be placed so that it covers a non-zero total, and the same for a (6m + 4) × (6n + 3) rectangle. Thus these cannot be tiled by the D hexomino. The only remaining possibilities with area divisible by 6 have one side divisible by 6. QED.

Theorem. Any rectangle tiled by the D hexomino has one side divisible by 4 .

This is a more difficult result. It is stated in [1, Theorem 7.1] and proved in [2, Theorem 5.12].

References

[1] Michael Reid, Tile Homotopy Groups, L'Enseignement Mathématique 49 (2003), no. 1-2, pp. 123-155.
[2] Michael Reid, Klarner Systems and Tiling Boxes with Polyominoes, Journal of Combinatorial Theory, Series A 111 (2005), no. 1, pp. 89-105.

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Updated August 25, 2011.