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

smallest odd rectangle: 27 × 29

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.