Primes of the dekomino

[dekomino]

18 × 60, 18 × 65, 18 × 70, 18 × 75, 18 × 80, 18 × 85, 18 × 90, 18 × 95, 18 × 100, 18 × 105, 18 × 110, 18 × 115
20 × 54, 20 × 59, 20 × 60, 20 × 65, 20 × 70, 20 × 72, 20 × 73, 20 × 74, 20 × 75, 20 × 77, 20 × 78, 20 × 79, 20 × 80, 20 × 81, 20 × 82, 20 × 83, 20 × 84, 20 × 85, 20 × 86, 20 × 87, 20 × 88, 20 × 89, 20 × 90, 20 × 91, 20 × 92, 20 × 93, 20 × 94, 20 × 95, 20 × 96, 20 × 97, 20 × 98, 20 × 99, 20 × 100, 20 × 101, 20 × 102, 20 × 103, 20 × 104, 20 × 105, 20 × 106, 20 × 107, 20 × 109, 20 × 110, 20 × 111, 20 × 112, 20 × 115, 20 × 116, 20 × 117, 20 × 121, 20 × 122, 20 × 123
22 × 60, 22 × 65, 22 × 70, 22 × 75, 22 × 80, 22 × 85, 22 × 90, 22 × 95, 22 × 100, 22 × 105, 22 × 110, 22 × 115
25 × 44, 25 × 60, 25 × 76, 25 × 80, 25 × 82, 25 × 84, 25 × 90, 25 × 92, 25 × 94, 25 × 96, 25 × 98, 25 × 100, 25 × 102, 25 × 106, 25 × 108, 25 × 110, 25 × 112, 25 × 114, 25 × 116, 25 × 118, 25 × 122, 25 × 130
27 × 60, 27 × 70, 27 × 80, 27 × 90, 27 × 100, 27 × 110
28 × 55, 28 × 60, 28 × 65, 28 × 70, 28 × 75, 28 × 80, 28 × 85, 28 × 90, 28 × 95, 28 × 100, 28 × 105
30 × 32, 30 × 40, 30 × 44, 30 × 46, 30 × 47, 30 × 48, 30 × 49, 30 × 50, 30 × 51, 30 × 52, 30 × 53, 30 × 54, 30 × 55, 30 × 56, 30 × 57, 30 × 58, 30 × 59, 30 × 60, 30 × 61, 30 × 62, 30 × 63, 30 × 65, 30 × 66, 30 × 67, 30 × 68, 30 × 69, 30 × 70, 30 × 71, 30 × 73, 30 × 74, 30 × 75, 30 × 77
31 × 60, 31 × 70, 31 × 80, 31 × 90, 31 × 100, 31 × 110
32 × 40, 32 × 45, 32 × 50, 32 × 55, 32 × 65
33 × 40, 33 × 60, 33 × 70, 33 × 90
34 × 40, 34 × 45, 34 × 50, 34 × 55, 34 × 60, 34 × 65, 34 × 70, 34 × 75
35 × 36, 35 × 40, 35 × 42, 35 × 44, 35 × 46, 35 × 48, 35 × 50, 35 × 52, 35 × 54, 35 × 56, 35 × 58, 35 × 60, 35 × 62, 35 × 64, 35 × 66, 35 × 68, 35 × 70, 35 × 74
36 × 40, 36 × 45, 36 × 50, 36 × 55
37 × 40, 37 × 50, 37 × 60, 37 × 70
38 × 40, 38 × 45, 38 × 50, 38 × 55
39 × 40, 39 × 50, 39 × 60, 39 × 70
40 × 40, 40 × 41, 40 × 42, 40 × 43, 40 × 44, 40 × 45, 40 × 46, 40 × 47, 40 × 48, 40 × 49, 40 × 50, 40 × 51, 40 × 52, 40 × 53, 40 × 55, 40 × 56, 40 × 57, 40 × 58, 40 × 61
41 × 50, 41 × 60, 41 × 70
42 × 45, 42 × 50, 42 × 55
43 × 50, 43 × 70
44 × 45
45 × 46, 45 × 48, 45 × 50, 45 × 52, 45 × 54, 45 × 56, 45 × 58, 45 × 62
46 × 50, 46 × 55
47 × 50
48 × 50, 48 × 55
49 × 50
50 × 50, 50 × 51, 50 × 52, 50 × 53, 50 × 55, 50 × 56, 50 × 57, 50 × 58, 50 × 61, 50 × 63
52 × 55
complete


smallest rectangle: 30 × 32

[30 x 32 rectangle]


smallest odd rectangle: 18 × 65

[18 x 65 rectangle]


The smallest rectangle for this shape was found by William Rex Marshall (see [1, Figure 153]). It has no symmetric tiling. The smallest odd rectangle was given in [2, Figure 11].

If A, B ≥ 36 and AB is a multiple of 10, then this dekomino tiles an A × B rectangle. The condition that AB is a multiple of 10 is certainly necessary. The bound 36 cannot be lowered, since it does not tile a 35 × 38 rectangle. See [3, Example 4.15].


References

[1] Solomon W. Golomb, Polyominoes, Second edition, Princeton University Press, 1994.
[2] Michael Reid, Tiling Rectangles and Half Strips with Congruent Polyominoes, Journal of Combinatorial Theory, Series A 80 (1997), no. 1, pp. 106-123.
[3] Michael Reid, Asymptotically Optimal Box Packing Theorems, The Electronic Journal of Combinatorics 15 (2008), no. 1, R78, 19 pp.


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Updated May 25, 2011.