Primes of the 11-omino

[11-omino]

30 × 154, 30 × 165, 30 × 176, 30 × 187, 30 × 198, 30 × 209, 30 × 220, 30 × 231, 30 × 242, 30 × 253, 30 × 264, 30 × 275, 30 × 286, 30 × 297
36 × 187, 36 × 198, 36 × 209, 36 × 220, 36 × 231, 36 × 242, 36 × 253, 36 × 264, 36 × 275, 36 × 286, 36 × 297, 36 × 308, 36 × 319, 36 × 330, 36 × 341, 36 × 352, 36 × 363
39 × 132, 39 × 143, 39 × 154, 39 × 165, 39 × 176, 39 × 187, 39 × 198, 39 × 209, 39 × 220, 39 × 231, 39 × 242, 39 × 253
41 × 132, 41 × 143, 41 × 154, 41 × 165, 41 × 176, 41 × 187, 41 × 198, 41 × 209, 41 × 220, 41 × 231, 41 × 242, 41 × 253
42 × 154, 42 × 165, 42 × 176, 42 × 187, 42 × 198, 42 × 209, 42 × 220, 42 × 231, 42 × 242, 42 × 253, 42 × 264, 42 × 275, 42 × 286, 42 × 297
43 × 110, 43 × 121, 43 × 132, 43 × 143, 43 × 154, 43 × 165, 43 × 176, 43 × 187, 43 × 198, 43 × 209
44 × 134, 44 × 138, 44 × 140, 44 × 142, 44 × 144, 44 × 145, 44 × 150, 44 × 151, 44 × 154, 44 × 155, 44 × 156, 44 × 157, 44 × 158, 44 × 159, 44 × 160, 44 × 161, 44 × 162, 44 × 163, 44 × 164, 44 × 165, 44 × 166, 44 × 167, 44 × 168, 44 × 169, 44 × 170, 44 × 171, 44 × 172, 44 × 173, 44 × 174, 44 × 175, 44 × 176, 44 × 177, 44 × 178, 44 × 179, 44 × 180, 44 × 181, 44 × 182, 44 × 183, 44 × 184, 44 × 185, 44 × 186, 44 × 187, 44 × 188, 44 × 189, 44 × 190, 44 × 191, 44 × 192, 44 × 193, 44 × 194, 44 × 195, 44 × 196, 44 × 197, 44 × 198, 44 × 199, 44 × 200, 44 × 201, 44 × 202, 44 × 203, 44 × 204, 44 × 205, 44 × 206, 44 × 207, 44 × 208, 44 × 209, 44 × 210, 44 × 211, 44 × 212, 44 × 213, 44 × 214, 44 × 215, 44 × 216, 44 × 217, 44 × 218, 44 × 219, 44 × 220, 44 × 221, 44 × 222, 44 × 223, 44 × 224, 44 × 225, 44 × 226, 44 × 227, 44 × 228, 44 × 229, 44 × 230, 44 × 231, 44 × 232, 44 × 233, 44 × 234, 44 × 235, 44 × 236, 44 × 237, 44 × 238, 44 × 239, 44 × 240, 44 × 241, 44 × 242, 44 × 243, 44 × 244, 44 × 245, 44 × 246, 44 × 247, 44 × 248, 44 × 249, 44 × 250, 44 × 251, 44 × 252, 44 × 253, 44 × 254, 44 × 255, 44 × 256, 44 × 257, 44 × 258, 44 × 259, 44 × 260, 44 × 261, 44 × 262, 44 × 263, 44 × 264, 44 × 265, 44 × 266, 44 × 267, 44 × 269, 44 × 270, 44 × 271, 44 × 273, 44 × 275, 44 × 277, 44 × 281
45 × 88, 45 × 110, 45 × 121, 45 × 132, 45 × 143, 45 × 154, 45 × 165, 45 × 187
46 × 110, ...
47 × 99, ...
48 × 110, 48 × 121, 48 × 132, 48 × 143, 48 × 154, 48 × 165, 48 × 176, 48 × 187, 48 × 198, 48 × 209
54 × 55, 54 × 66, 54 × 77, 54 × 88, 54 × 99
...


smallest rectangle: 54 × 55

[54 x 55 rectangle]


smallest known odd rectangle: 47 × 99

[47 x 99 rectangle]


A 30 × 154 rectangle is given in [2, Figure 8], and Marshall [1, Figure 10] gives a larger (44 × 208) rectangle. The 54 × 55 rectangle above is the smallest. This is the first known case where the rectangular order (270) is congruent to 6 mod 8.


References

[1] William Rex Marshall, Packing Rectangles with Congruent Polyominoes, Journal of Combinatorial Theory, Series A 77 (1997), no. 2, pp. 181-192.
[2] Michael Reid, Tiling Rectangles and Half Strips with Congruent Polyominoes, Journal of Combinatorial Theory, Series A 80 (1997), no. 1, pp. 106-123.


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Updated August 25, 2011.