Primes of the octomino

[octomino]

26 × 96, 26 × 104, 26 × 112, 26 × 120, 26 × 128, 26 × 136, 26 × 144, 26 × 152, 26 × 160, 26 × 168, 26 × 176, 26 × 184
28 × 128, 28 × 132, 28 × 136, 28 × 140, 28 × 144, 28 × 148, 28 × 152, 28 × 156, 28 × 160, 28 × 164, 28 × 168, 28 × 172, 28 × 176, 28 × 180, 28 × 184, 28 × 188, 28 × 192, 28 × 196, 28 × 200, 28 × 204, 28 × 208, 28 × 212, 28 × 216, 28 × 220, 28 × 224, 28 × 228, 28 × 232, 28 × 236, 28 × 240, 28 × 244, 28 × 248, 28 × 252
29 × 208, 29 × 224, 29 × 240, 29 × 256, 29 × 272, 29 × 288, 29 × 304, 29 × 320, 29 × 336, 29 × 352, 29 × 368, 29 × 384, 29 × 400
30 × 80, 30 × 88, 30 × 96, 30 × 104, 30 × 112, 30 × 120, 30 × 128, 30 × 136, 30 × 144, 30 × 152
31 × 128, 31 × 144, 31 × 160, 31 × 176, 31 × 192, 31 × 208, 31 × 224, 31 × 240
32 × 74, 32 × 78, 32 × 82, 32 × 83, 32 × 84, 32 × 86, 32 × 88, 32 × 90, 32 × 92, 32 × 93, 32 × 94, 32 × 96, 32 × 97, 32 × 98, 32 × 99, 32 × 100, 32 × 101, 32 × 102, 32 × 103, 32 × 104, 32 × 105, 32 × 106, 32 × 107, 32 × 108, 32 × 109, 32 × 110, 32 × 111, 32 × 112, 32 × 113, 32 × 114, 32 × 115, 32 × 116, 32 × 117, 32 × 118, 32 × 119, 32 × 120, 32 × 121, 32 × 122, 32 × 123, 32 × 124, 32 × 125, 32 × 126, 32 × 127, 32 × 128, 32 × 129, 32 × 130, 32 × 131, 32 × 132, 32 × 133, 32 × 134, 32 × 135, 32 × 136, 32 × 137, 32 × 138, 32 × 139, 32 × 140, 32 × 141, 32 × 142, 32 × 143, 32 × 144, 32 × 145, 32 × 146, 32 × 147, 32 × 149, 32 × 150, 32 × 151, 32 × 153, 32 × 154, 32 × 155, 32 × 159, 32 × 163
33 × 112, 33 × 128, 33 × 144, ...
34 × 64, 34 × 80, 34 × 96, 34 × 112, ...
35 × 112, ...
37 × 64, ...
41 × 48, ...
45 × 48, ...
48 × 48, 48 × 49, 48 × 53, 48 × 57, 48 × 61, 48 × 65, 48 × 69, 48 × 73, 48 × 77, 48 × 81, ...
...


smallest rectangle: 41 × 48

[41 x 48 rectangle]


The first rectangle for this octomino, 26 × 96 , was found independently by Karl Dahlke [1, Figure 1] (see also [2, Figure 154]) and Wrede [3, Figure 5.3.8]. This is the second known polyomino, after the P 11-omino, whose rectangular order, 246, is congruent to 6 mod 8.


References

[1] Karl A. Dahlke, Solomon W. Golomb and Herbert Taylor, An Octomino of High Order, Journal of Combinatorial Theory, Series A 70 (1995) pp. 157-158.
[2] Solomon W. Golomb, Polyominoes, Second edition, Princeton University Press, 1994.
[3] Ingo Wrede, Rechteckzerlegungen mit kleinen Polyominos, 1990 Diplomarbeit, Technische Universität Braunschweig, (unpublished).


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Updated May 19, 2012.