Primes of the A hexomino

3 × 4
complete

smallest rectangle: 3 × 4

Proposition. Any rectangle tiled by the A hexomino has one side divisible by 4.
Proof. It suffices to show that it doesn't tile any (4m + 2) × (4n + 2) rectangle. Consider the numbering

          (x, y) |---> { 1  if  x  and  y  are both even
                       { 0  otherwise.
  

No matter how it is placed, each A hexomino covers an odd total. Also, a (4m + 2) × (4n + 2) rectangle covers an odd total. However, it would be tiled by an even number of A hexominoes, which cover an even total, a contradiction. QED.

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Updated May 25, 2005.