Primes of the heptomino

13 × 112, 13 × 126, 13 × 140, 13 × 147, 13 × 154, 13 × 161, 13 × 168, 13 × 175, 13 × 182, 13 × 189, 13 × 196, 13 × 203, 13 × 210, 13 × 217, 13 × 231, 13 × 245
18 × 35, 18 × 42, 18 × 49, 18 × 56, 18 × 63
19 × 28, 19 × 42, 19 × 77, 19 × 91
20 × 49, 20 × 56, 20 × 63, 20 × 70, 20 × 77, 20 × 84, 20 × 91
21 × 26, 21 × 32, 21 × 38, 21 × 40, 21 × 44, 21 × 46, 21 × 50, 21 × 51, 21 × 54, 21 × 56, 21 × 57, 21 × 59, 21 × 60, 21 × 61, 21 × 62, 21 × 63, 21 × 65, 21 × 67, 21 × 68, 21 × 69, 21 × 71, 21 × 73, 21 × 74, 21 × 75, 21 × 79, 21 × 81
24 × 35, 24 × 42, 24 × 49, 24 × 56, 24 × 63
25 × 28, 25 × 42, 25 × 49, 25 × 63
26 × 28, 26 × 35
27 × 42, 27 × 49, 27 × 56, 27 × 63, 27 × 70, 27 × 77
28 × 31, 28 × 32, 28 × 33, 28 × 37, 28 × 39, 28 × 40, 28 × 42, 28 × 43, 28 × 46, 28 × 47, 28 × 48, 28 × 49, 28 × 53, 28 × 54, 28 × 55, 28 × 60
30 × 35, 30 × 42, 30 × 49, 30 × 56, 30 × 63
31 × 35, 31 × 42, 31 × 49
32 × 35
33 × 35, 33 × 42, 33 × 49
34 × 42, 34 × 49, 34 × 56, 34 × 63, 34 × 70, 34 × 77
35 × 37, 35 × 38, 35 × 39, 35 × 40, 35 × 43, 35 × 45, 35 × 46, 35 × 47, 35 × 53
37 × 49
39 × 42, 39 × 49
41 × 42, 41 × 49
complete

smallest rectangle: 19 × 28




smallest odd rectangle: 21 × 51

Karl Scherer [6a, 6b] was the first to find a rectangle (26 × 42) for this shape. Marlow [3] and Karl Dalhke [1a, 1b, 6c] subsequently found the minimal rectangle, 19 × 28. As noted by Dahlke [1a], this is the first example where the minimal rectangle has no symmetric tiling. This settles a question of Klarner [2] (although Dahlke did not mention this). The smallest odd rectangle is given in [4, Figure 11].

If a, b ≥ 36 and 7 divides ab, then this heptomino tiles an a × b rectangle. The condition that 7 divides ab is necessary, because the area must be divisible by 7. Moreover, the bound 36 cannot be lowered, since it does not tile a 35 × 41 rectangle. Also see [5, Example 4.14].

References

[1a] Karl A. Dahlke, A Heptomino of Order 76, Journal of Combinatorial Theory, Series A, 51 (1989), pp. 127-128.
[1b] Erratum, Journal of Combinatorial Theory, Series A, 52 (1990), p. 321.
[2] David A. Klarner, Some Results Concerning Polyominoes, Fibonacci Quarterly 3 (1965), no. 1, pp. 9-20.
[3] T.W. Marlow, Grid Dissections, Chessics 23 (1985), pp. 78-79.
[4] Michael Reid, Tiling Rectangles and Half Strips with Congruent Polyominoes, Journal of Combinatorial Theory, Series A 80 (1997), no. 1, pp. 106-123.
[5] Michael Reid, Asymptotically Optimal Box Packing Theorems, The Electronic Journal of Combinatorics 15 (2008), no. 1, R78, 19 pp.
[6a] Karl Scherer, Heptomino Tessellations, Problem 1045, Journal of Recreational Mathematics 14 (1981-1982) p. 64.
[6b] Solutions by Scherer and Karl A. Dahlke, Journal of Recreational Mathematics 21 (1989) pp. 221-223.
[6c] Solution by Karl A. Dahlke, Journal of Recreational Mathematics 22 (1990) pp. 68-69.

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Updated December 27, 2018.