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.