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 × 42, ...

34 × 42, 34 × 49, 34 × 56, 34 × 63, ...

35 × 38, 35 × 40, 35 × 45, 35 × 46, ...

39 × 42, 39 × 49

41 × 42, 41 × 49

smallest rectangle: 19 × 28

smallest odd rectangle: 21 × 51

Karl Scherer [5a, 5b] was the first to find a rectangle (26 × 42) for this shape. Marlow [3] and Karl Dalhke [1a, 1b] 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].

**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.

[5a] Karl Scherer,
Heptomino Tessellations, Problem 1045,
*Journal of Recreational Mathematics* **14** (1981-1982)
p. 64.

[5b] Solutions by Scherer
and Karl A.
Dahlke,
*Journal of Recreational Mathematics* **21** (1989)
pp. 221-223.

[5c] Solution by Karl
A. Dahlke, *Journal of Recreational Mathematics* **22**
(1990) pp. 68-69.

Data for prime rectangles | Rectifiable polyominoes | Polyomino page | Home page | E-mail

Updated August 23, 2011.