Primes of the 12-omino

[12-omino]

26 × 216, 26 × 240, 26 × 264, 26 × 288, 26 × 312, 26 × 336, 26 × 360, 26 × 384, 26 × 408
27 × 368, 27 × 376, 27 × 384, 27 × 392, 27 × 400, 27 × 408, 27 × 416, 27 × 424, 27 × 432, 27 × 440, 27 × 448, 27 × 456, 27 × 464, 27 × 472, 27 × 480, 27 × 488, 27 × 496, 27 × 504, 27 × 512, 27 × 520, 27 × 528, 27 × 536, 27 × 544, 27 × 552, 27 × 560, 27 × 568, 27 × 576, 27 × 584, 27 × 592, 27 × 600, 27 × 608, 27 × 616, 27 × 624, 27 × 632, 27 × 640, 27 × 648, 27 × 656, 27 × 664, 27 × 672, 27 × 680, 27 × 688, 27 × 696, 27 × 704, 27 × 712, 27 × 720, 27 × 728
28 × 132, 28 × 156, 28 × 168, 28 × 180, 28 × 192, 28 × 204, 28 × 216, 28 × 228, 28 × 240, 28 × 252, 28 × 276
29 × 144, 29 × 168, 29 × 192, 29 × 216, 29 × 240, 29 × 264
30 × 128, 30 × 136, 30 × 144, 30 × 152, 30 × 160, 30 × 168, 30 × 176, 30 × 184, 30 × 192, 30 × 200, 30 × 208, 30 × 216, 30 × 224, 30 × 232, 30 × 240, 30 × 248
31 × 120, 31 × 144, 31 × 168, 31 × 192, 31 × 216
32 × 129, 32 × 135, 32 × 138, 32 × 141, 32 × 144, 32 × 147, 32 × 150, 32 × 153, 32 × 156, 32 × 159, 32 × 162, 32 × 165, 32 × 168, 32 × 171, 32 × 174, 32 × 177, 32 × 180, 32 × 183, 32 × 186, 32 × 189, 32 × 192, 32 × 195, 32 × 198, 32 × 201, 32 × 204, 32 × 207, 32 × 210, 32 × 213, 32 × 216, 32 × 219, 32 × 222, 32 × 225, 32 × 228, 32 × 231, 32 × 234, 32 × 237, 32 × 240, 32 × 243, 32 × 246, 32 × 249, 32 × 252, 32 × 255, 32 × 261
33 × 128, 33 × 136, 33 × 144, 33 × 152, 33 × 160, 33 × 168, 33 × 176, 33 × 184, 33 × 192, 33 × 200, 33 × 208, 33 × 216, 33 × 224, 33 × 232, 33 × 240, 33 × 248
34 × 120, 34 × 144, 34 × 168, 34 × 192, 34 × 216
35 × 96, 35 × 120, 35 × 144, 35 × 168
36 × 88, 36 × 92, 36 × 96, 36 × 100, 36 × 104, 36 × 108, 36 × 112, 36 × 116, 36 × 120, 36 × 124, 36 × 128, 36 × 132, 36 × 136, 36 × 140, 36 × 144, 36 × 148, 36 × 152, 36 × 156, 36 × 160, 36 × 164, 36 × 168, 36 × 172
37 × 96, 37 × 120, 37 × 144, 37 × 168
38 × 72, 38 × 96, 38 × 120
39 × 64, 39 × 72, 39 × 80, 39 × 88, 39 × 96, 39 × 104, 39 × 112, 39 × 120
40 × 72, 40 × 75, 40 × 78, 40 × 81, 40 × 84, 40 × 87, 40 × 90, 40 × 93, 40 × 96, 40 × 99, 40 × 102, 40 × 105, 40 × 108, 40 × 111, 40 × 114, 40 × 117, 40 × 120, 40 × 123, 40 × 126, 40 × 129, 40 × 132, 40 × 135, 40 × 138, 40 × 141
41 × 96, 41 × 120, 41 × 144, 41 × 168
42 × 64, 42 × 88, 42 × 112, ...
43 × 72, 43 × 96, 43 × 120
44 × 60, 44 × 84, 44 × 96, 44 × 108, ...
45 × 48, 45 × 56, 45 × 64, 45 × 72, 45 × 80, 45 × 88
48 × 49, 48 × 51, 48 × 53, 48 × 55, 48 × 57, 48 × 59, 48 × 61, 48 × 63, 48 × 65, 48 × 67, 48 × 69, 48 × 71, 48 × 73, 48 × 75, 48 × 77, 48 × 79, 48 × 81, 48 × 83, 48 × 85, 48 × 87, 48 × 89, ...
49 × 72
51 × 56, 51 × 64, 51 × 72, 51 × 80, 51 × 88
53 × 72
55 × 72
56 × 57, 56 × 63, 56 × 69, 56 × 75, 56 × 81, 56 × 87, ...
57 × 64, 57 × 72, 57 × 80, 57 × 88
59 × 72
60 × 68, ...
61 × 72
62 × 72, ...
63 × 64, 63 × 72, 63 × 80, 63 × 88
64 × 66, 64 × 69, 64 × 72, 64 × 75
65 × 72
66 × 88, ...
67 × 72
68 × 84, ...
69 × 72, 69 × 80, 69 × 88
71 × 72
72 × 72, 72 × 73, 72 × 75, ...
...


smallest rectangle: 45 × 48

[45 x 48 rectangle]


A 28 × 132 rectangle was given in [1, Figure 4]. The 45 × 48 rectangle shown above is the smallest rectangle.

In [1, Proposition 3.5] it is proved that this polyomino is even. The following stronger condition is proved in [2, Theorem 4.5].

Theorem. If an m × n rectangle can be tiled by this shape, then either m is a multiple of 4 or n is a multiple of 8.


References

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


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Updated September 16, 2011.