Primes of the P heptomino

11 × 35, 11 × 42, 11 × 49, 11 × 56, 11 × 63
12 × 21, 12 × 28, 12 × 35
14 × 14, 14 × 19, 14 × 20, 14 × 22, 14 × 23, 14 × 24, 14 × 25, 14 × 26, 14 × 27, 14 × 29, 14 × 30, 14 × 31, 14 × 32, 14 × 35
15 × 21, 15 × 28, 15 × 35
16 × 21, 16 × 28, 16 × 35
17 × 21, 17 × 28, 17 × 35
18 × 21, 18 × 28, 18 × 35
19 × 21
20 × 21
21 × 21, 21 × 22, 21 × 23, 21 × 25, 21 × 26
complete

smallest rectangle: 14 × 14




smallest odd rectangle: 15 × 21

Klarner [2, Figure 4] gives the first rectangle, 12 × 21, for this shape, and later [3, Figure 6] gives the minimal rectangle, 14 × 14. Gardner [1, Figure 83, p. 187] also gives the 14 × 14 square, and attributes it to James E. Stuart. The smallest odd rectangle, 15 × 21, was given in [4, Figure 13] and [5, Figure 11]. All the prime rectangles were also found independently by Andrew Clarke.

References

[1] Martin Gardner, Polyominoes and Rectification, Chapter 13 in Mathematical Magic Show, The Mathematical Association of America, 1989.
[2] David A. Klarner, Some Results Concerning Polyominoes, Fibonacci Quarterly 3 (1965), no. 1, pp. 9-20.
[3] David A. Klarner, Packing a Rectangle with Congruent N-ominoes, Journal of Combinatorial Theory 7 (1969) pp. 107-115.
[4] William Rex Marshall, Packing Rectangles with Congruent Polyominoes, Journal of Combinatorial Theory, Series A 77 (1997), no. 2, pp. 181-192.
[5] 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.