Primes of the Y hexomino

23 × 24
24 × 29, 24 × 35, 24 × 41, 24 × 47, 24 × 53, 24 × 59, 24 × 63, 24 × 65, 24 × 71, 24 × 77, 24 × 83, 24 × 89, 24 × 95, 24 × 101, 24 × 102, 24 × 103, 24 × 107, 24 × 108, 24 × 113, 24 × 114, 24 × 119, 24 × 120
30 × 64, 30 × 68, 30 × 72, 30 × 80, 30 × 88, 30 × 92, 30 × 96, 30 × 100, 30 × 104, 30 × 106, 30 × 108, 30 × 112, 30 × 116, 30 × 120, 30 × 124, ...
32 × 36, 32 × 42, 32 × 48, 32 × 54, 32 × 60, 32 × 66, ...
48 × 48, ...
...

smallest rectangle: 23 × 24

The minimal 23 × 24 rectangle was found independently by Karl A. Dahlke [1], and T.W. Marlow [2].

Proposition. If the Y hexomino tiles a rectangle with an odd side, then the other side is divisible by 8.
Proof. It suffices to show that it does not tile any (2m + 1) × (8n + 4) rectangles. Consider the numbering

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

No matter how it is placed, each Y hexomino covers a total of 2 or -2 , hence, 2 mod 4. A (2m + 1) × (8n + 4) also covers a total that is 2 mod 4. However, it would be tiled by an even number of Y hexominoes, which would cover a total that is divisible by 4 , a contradiction. QED.

References

[1] Karl A. Dahlke, The Y-hexomino has order 92, Journal of Combinatorial Theory, Series A 51 (1989), pp. 125-126.
[2] T.W. Marlow, Grid Dissections, Chessics 23 (1985), pp. 78-79.

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Updated August 25, 2011.