Primes of the 11-omino

22 × 25, 22 × 28, 22 × 34, 22 × 40, 22 × 46, 22 × 48, 22 × 52, 22 × 54, 22 × 58, 22 × 60, 22 × 64, 22 × 66, 22 × 70, 22 × 72
25 × 187
28 × 77
30 × 77, 30 × 88, 30 × 99, 30 × 110, 30 × 121, 30 × 132, 30 × 143
31 × 44, 31 × 99, 31 × 110, 31 × 121
33 × 42, 33 × 46, 33 × 47, 33 × 48, 33 × 50, 33 × 51, 33 × 53, 33 × 54, 33 × 56, 33 × 58, 33 × 59, 33 × 61, 33 × 62, 33 × 64, 33 × 65, 33 × 66, 33 × 67, 33 × 68, 33 × 69, 33 × 70, 33 × 71, 33 × 72, 33 × 73, 33 × 74, 33 × 75, 33 × 76, 33 × 77, 33 × 78, 33 × 79, 33 × 80, 33 × 81, 33 × 82, 33 × 83, 33 × 85, 33 × 86, 33 × 87, 33 × 91
34 × 77
36 × 44, 36 × 55, 36 × 66, 36 × 77
37 × 66, 37 × 77, 37 × 88, 37 × 99, 37 × 110, 37 × 121
38 × 66, 38 × 77, 38 × 88, 38 × 99, 38 × 110, 38 × 121
39 × 44, 39 × 55, 39 × 66, 39 × 77
40 × 55
41 × 66, 41 × 77, 41 × 88, 41 × 99, 41 × 110, 41 × 121
42 × 44, 42 × 55
43 × 66, 43 × 77, 43 × 88, 43 × 99, 43 × 110, 43 × 121
44 × 44, 44 × 45, 44 × 47, 44 × 51, 44 × 55, 44 × 57, 44 × 63
45 × 55, 45 × 66, 45 × 77
47 × 55
49 × 55, 49 × 66, 49 × 77, 49 × 88, 49 × 99
51 × 55
52 × 55
55 × 55, 55 × 57, 55 × 60, 55 × 61, 55 × 63, 55 × 67, 55 × 69
57 × 66, 57 × 77
complete

smallest rectangle: 22 × 25




smallest odd rectangle: 33 × 47

The smallest rectangle was found independently by Marshall (see [1, Figure 151]) and Wrede [2, Figure 5.6.4]. Marshall also found the smallest odd rectangle (see [1, Figure 163]). Both rectangles have unique tilings. This is the only known polyomino (other than the trivial case of rectangles) for which the smallest odd rectangle has a unique tiling.

If A, B ≥ 45 and AB is a multiple of 11, then this polyomino tiles an A × B rectangle. The condition that AB is a multiple of 11 is required because the area must be divisible by 11. The value 45 cannot be lowered, since it does not tile a 44 × 49 rectangle.

References

[1] Solomon W. Golomb, Polyominoes, Second edition, Princeton University Press, 1994.
[2] Ingo Wrede, Rechteckzerlegungen mit kleinen Polyominos, 1990 Diplomarbeit, Technische Universität Braunschweig, (unpublished).

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Updated September 7, 2017.