Problem of the Month (January 2001)

Determining which polyominoes tile rectangles of what sizes is a well-known problem. See Mike Reid's Polyomino Page for some neat pictures.

This month we consider collections of two polyominoes, neither of which can tile a rectangle separately, and ask which rectangles they tile jointly. For example, the U pentomino and the S tetromino can tile a 3x6 rectangle:

For a less trivial example, the V pentomino and the S tetromino can tile a 8x18 rectangle:

What other rectangles can these pairs of polyominoes tile? What about other pairs of non-rectifiable polyominoes?


ANSWERS

Turns out this has been well studied. All this data comes from Mike Reid, though there were also contributions from Joseph DeVincentis, Torsten Sillke, and Patrick Hamlyn.

Here are the smallest rectangles, and a list of prime rectangles for pairs of tetrominoes and pentominoes:

Prime Rectangles:
5x14, 5x20, 5x26, 5x32, 5x38, 5x44, 5x50
6x6, 6x8, 6x10, 6x11, 6x13, 6x15
7x10, 7x12, 7x13, 7x14, 7x16, 7x17, 7x18, 7x19, 7x21
8x8, 8x10, 8x11, 8x13, 8x15
9x10, 9x11, 9x12, 9x13, 9x14, 9x15, 9x16, 9x17, 9x18, 9x19
10x10, 10x11
11x11, 11x13
Prime Rectangles:
3x6
5x10, 5x18, 5x22, 5x26, 5x34
6x16, 6x17
7x10, 7x16, 7x18, 7x22, 7x23, 7x24, 7x25, 7x27, 7x29, 7x31
8x11, 8x12, 8x13, 8x14, 8x15, 8x16, 8x17, 8x19, 8x20, 8x21
9x10, 9x11, 9x13, 9x14, 9x15
10x11, 10x13
11x11, 11x13, 11x14, 11x15
13x13, 13x14, 13x15
14x14
Prime Rectangles:
8x16, 8x18, 8x20, 8x22, 8x24, 8x26, 8x28, 8x30
9x18, 9x21, 9x24, 9x27, 9x30, 9x33
10x14, 10x16, 10x18, 10x20, 10x22, 10x24, 10x26
11x12, 11x18
12x12, 12x13, 12x14, 12x15, 12x16, 
12x17, 12x18, 12x19, 12x20, 12x21
13x18
14x14, 14x16, 14x18
15x15, 15x18, 15x21
Prime Rectangles:
3x10
6x15
7x10, 7x15
8x25, 8x35, 8x40, 8x45, 8x55
9x15
10x11
11x15
Prime Rectangles:
4x5
6x10
7x20, 7x30, 7x45, 7x55
9x10, 9x15
10x11
11x15
Prime Rectangles:
5x10, 5x16, 5x18, 5x22, 5x24
6x10, 6x15
7x10, 7x15
8x10, 8x15
9x10, 9x15
11x15
Prime Rectangles:
4x5
7x40, 7x50, 7x55, 7x60, 7x65, 7x70, 7x75, 7x85
9x15, 9x25
10x13, 10x14, 10x15
11x15, 11x20, 11x25
Prime Rectangles:
4x5
5x6
7x10, 7x15
9x15
11x15
Prime Rectangles:
12x30, 12x40, 12x45, 12x50, 12x55, 12x65
13x25, 13x30, 13x35, 13x40, 13x45
14x35, 14x40, 14x45, 14x50, 14x55, 14x60, 14x65
15x24, 15x28, 15x29, 15x31, 15x32, 15x33, 15x34, 15x35, 
15x36, 15x37, 15x38, 15x39, 15x40, 15x41, 15x42, 15x43, 
15x44, 15x45, 15x46, 15x47, 15x49, 15x50, 15x51
16x30, 16x35, 16x40, 16x45, 16x50, 16x55
17x20, 17x25, 17x30, 17x35
18x20, 18x25, 18x30, 18x35
19x25, 19x30, 19x35, 19x40, 19x45
20x20, 20x21, 20x22, 20x23, 20x24, 20x25, 20x26, 
20x27, 20x28, 20x29, 20x30, 20x31, 20x32, 20x33
21x25, 21x30, 21x35
22x25, 22x30
Prime Rectangles:
8x10, 8x15
9x10, 9x25
10x10, 10x11, 10x12, 10x13, 10x14, 10x15
11x15
12x15
13x15
14x15
15x15, 15x17
Prime Rectangles:
14x40, 14x45, 14x50, 14x55, 14x60, 14x65, 14x70, 14x75
15x28, 15x32, 15x36, 15x37, 15x38, 15x40, 15x41, 15x42, 15x43, 15x44, 
15x45, 15x46, 15x47, 15x48, 15x49, 15x50, 15x51, 15x52, 15x53, 15x54, 
15x55, 15x57, 15x58, 15x59, 15x61, 15x62, 15x63, 15x67
16x30, 16x35, 16x40, 16x45, 16x50, 16x55
17x30, 17x35, 17x40, 17x45, 17x50, 17x55
18x30, 18x35, 18x40, 18x45, 18x50, 18x55
19x30, 19x35, 19x40, 19x45, 19x50, 19x55
20x20, 20x21, 20x22, 20x23, 20x25, 20x26, 20x27, 20x28, 20x29, 20x30, 
20x31, 20x32, 20x33, 20x34, 20x35, 20x36, 20x37, 20x38, 20x39
21x25, 21x30, 21x35
22x25, 22x30, 22x35
23x25, 23x30, 23x35
24x25, 24x30, 24x35, 24x40, 24x45
25x25, 25x26, 25x27, 25x28, 25x29, 25x30, 25x31, 25x32, 
25x33, 25x34, 25x35, 25x36, 25x37, 25x38, 25x39
26x30, 26x35
27x30, 27x35
29x30, 29x35
30x30, 30x31
Prime Rectangles:
3x5
17x85, 17x110
19x40, 19x65
20x70, 20x71
22x50
23x35, 23x40
25x28, 25x38
28x35
Prime Rectangles:
12x20, 12x25, 12x30, 12x35
13x35, 13x40, 13x45, 13x50, 13x55, 13x60, 13x65
14x30, 14x35, 14x40, 14x45, 14x50, 14x55
15x23, 15x24, 15x25, 15x26, 15x27, 15x28, 15x29, 15x30, 
15x31, 15x32, 15x33, 15x34, 15x35, 15x36, 15x37, 15x38, 
15x39, 15x40, 15x41, 15x42, 15x43, 15x44, 15x45
16x25, 16x30, 16x35, 16x40, 16x45
17x25, 17x30, 17x35, 17x40, 17x45
18x20, 18x25, 18x30, 18x35
19x20, 19x25, 19x30, 19x35
20x20, 20x21, 20x22, 20x23, 20x25, 20x26, 20x27, 20x28, 20x29
21x25, 21x30, 21x35
22x25, 22x30, 22x35
23x25
25x25, 25x26
Prime Rectangles:
3x5
4x5
Torsten Sillke and Mike Reid made me aware of the following web pages ( pentominoes, hexominoes, heptominoes ) and papers from the Journal of Recreational Mathematics:

Earl S. Kramer; Tiling Rectangles with T and C Pentominoes, 16 (1983) 102-113.

Earl S. Kramer, Frits Göbel; Tiling Rectangles with Pairs of Pentominoes, 16:3 (1983-84) 198-206.

Richard G. Laatsch; Rectangles from Mixed Polyomino Sets, 13:3 (1980-81) 183-187.

C. Jepsen; On tiling deficient rectangular boards with trominoes, (1995) v. 27(2) p. 125-130.

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 2/3/01.