Primes of the enneomino

6 × 6
18 × 25, 18 × 32, 18 × 34, 18 × 41, 18 × 45
19 × 72, 19 × 90, 19 × 108, 19 × 126
22 × 162, 22 × 180, 22 × 198, 22 × 216, 22 × 234, 22 × 252, 22 × 270, 22 × 288, 22 × 306
23 × 36, 23 × 54
24 × 33
26 × 54, 26 × 72, 26 × 90
27 × 48, 27 × 54, 27 × 60, 27 × 66, 27 × 72, 27 × 78, 27 × 84, 27 × 90, 27 × 93, 27 × 99, 27 × 105, 27 × 111, 27 × 117, 27 × 123, 27 × 129, 27 × 135
28 × 36, 28 × 54
30 × 39
33 × 36, 33 × 42, 33 × 63, 33 × 69, 33 × 75, 33 × 81
39 × 51, 39 × 57, 39 × 63, 39 × 69
45 × 45, 45 × 51, 45 × 57
51 × 51
complete

smallest rectangle: 6 × 6



smallest odd rectangle: 39 × 51

The smallest odd rectangle is given in [2, Figure 4.1].

The following result is proved in [1], see Example 4.7 and Theorem 4.8.

Theorem. If this enneomino tiles an a × b rectangle, then
(i) either 3 divides a or 18 divides b, and
(ii) either 18 divides a or 3 divides b.

If a, b ≥ 40 and satisfy the necessary conditions (i) and (ii) of the Theorem, then this shape tiles an a × b rectangle. Moreover, the bound 40 cannot be lowered, since it does not tile a 39 × 45 rectangle.

References

[1] Michael Reid, Asymptotical Optimal Box Packing Theorems, The Electronic Journal of Combinatorics 15 (2008), no. 1, R78, 19 pp.
[2] Michael Reid, Many L-Shaped Polyominoes Have Odd Rectangular Packings, Annals of Combinatorics 18 (2014), no. 2, pp. 341-357.

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Updated December 18, 2018.