Primes of the enneomino

[enneomino]

6 × 6
12 × 15
17 × 108, 17 × 180, 17 × 234, 17 × 243, 17 × 252, 17 × 261, 17 × 270, 17 × 279, 17 × 297, 17 × 306, 17 × 315, 17 × 333
18 × 52, 18 × 68, 18 × 69, 18 × 85, 18 × 95
21 × 69, 21 × 75, 21 × 78
22 × 27, 22 × 72, 22 × 90
23 × 36, 23 × 45, 23 × 54, 23 × 63
26 × 45, 26 × 54, 26 × 63, 26 × 72, 26 × 81
27 × 28, 27 × 29, 27 × 30, 27 × 35, 27 × 43, 27 × 45, 27 × 49
28 × 36, 28 × 45
29 × 45
31 × 45, 31 × 54, 31 × 63, 31 × 72, 31 × 81
32 × 36, 32 × 45, 32 × 63
33 × 33, 33 × 39
34 × 45
36 × 37
37 × 45, 37 × 63
39 × 39
complete


smallest rectangle: 6 × 6

[6 x 6 square]

smallest odd rectangle: 27 × 29

[27 x 29 rectangle]


The smallest odd rectangle was found by Marshall [1, Figure 13].

If A, B ≥ 32 and AB is a multiple of 9, then this shape tiles an A × B rectangle. The condition that AB is a multiple of 9 is certainly necessary. The bound 32 cannot be lowered, since it does not tile a 31 × 36 rectangle.


Reference

[1] William Rex Marshall, Packing Rectangles with Congruent Polyominoes, Journal of Combinatorial Theory, Series A 77 (1997), no. 2, pp. 181-192.


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Updated August 25, 2011.