Problem of the Month (September 2009)

As a designer of mathematical puzzles, I'm interested when a puzzle has a unique solution. This month we investigate when packing congruent polyominoes in a rectangle has a unique solution (up to rotations and reflections of the rectangle). We omit packings of 1 or 2 polyominoes as trivial. So which polyominoes and which rectangle sizes have unique packings? Can you prove that certain polyominoes only uniquely pack finitely many rectangles? What about packing rectangular cylinders, Möbius strips, tori, or Klein bottles? What if two polyominoes of the same area are used? What about unique packings of polyiamonds and polyhexes in roughly triangular or hexagonal shapes?

ANSWERS

Contributors this month include George Sicherman and Berend van der Zwaag.

Pentominoes in Rectangles
n \ m2345678
4 none none 3
5 none 3 4 4
5
6 none 3 none 5 none
7 none none 3
4 		4 none 6
8 none none 6 6 6
8 (BZ) 		7 11
12 (GS)
9 none 3 4 none 10 12 14 (GS)
10 4 6 6
8 		6 8 none 15 (BZ)
11 4 none 5 (GS) none 12 (GS) none none
12 none 4 none none 13 (BZ) 16 (BZ) 18 (BZ)

n \ m9101112
9 16 (GS)
10 17 (BZ) none
11 none 19 (BZ) 21 (BZ)
12 18 (BZ) 17 (BZ) none 27 (BZ)
28 (BZ)

Hexominoes in Rectangles
n \ m2345678
5 none none 3 (GS) 3
4

6 none 3 3
4 		4
5 		4
5
6
7 none 3 3
4

4 5
6 		6
8
8 none 3
4 		none 6 6 none 8
10 (all GS)
9 3 3
4 		4
5
6 		none 8 10 9 (GS)
11 (BZ)
10 3 4
5 		5
6

6 8
9 		10 (BZ)
11 (BZ) 		none
11 3 4
5 		5
6 		none 8 (GS)
9 (GS)
10 (GS) 		none 14 (all BZ)
12 4 5 (GS)
6 		7
8 		10 (all GS) 10 (GS)
11 (GS) 		none none

n \ m9101112
9 11 (BZ)
12 (BZ)
10 11 (all BZ) 14 (BZ)
16

(all BZ)

11 16 (BZ) 16 (BZ)
17 (BZ)
18 (BZ) 		16 (BZ)
17 (BZ)
20 (BZ)
12 16 (BZ)
18 (BZ) 		16 (BZ) none 20 (all BZ)

Heptominoes in Rectangles
n \ m345
5 none none 3
6 none 3 3
4
7 3 3

4 		3
4

5
8 none 3
4 		3
4
5
9 3

3
4

4
5
6
10 3 4
5 		4
5
6
11 3
4 		3
4
5
6 		5
6
7
12 3
4 		4
5
6 		6 (GS)
8 (GS)

n \ m678
6 3
4

7 4

5
6 		5
6
7
8 4
6 		6

7 		7
8

9 6 7
8 		8 (all GS)
9 (GS)
10 (GS)
10 6
7
8 		6 (GS)
7 (GS)
8 (all GS)
9 (GS) 		8 (GS)
9 (GS)
10 (all GS)
11 6 (GS)
8

(all GS)

8 (all GS)
9 (all GS) 		9 (GS)
10 (all GS)
12 9 (GS)
10 (GS) 		8 (GS)
10 (all GS) 		12 (GS)

Asymmetrical Unique Packings
Pentominoes 3 5 6 7 11 13

15 17 21 27

Hexominoes 3 5 6 7 9

10 11 14 16 17

Heptominoes 3

4 5

6 7

8 9 10

Berend van der Zwaag proved that the P-hexomino only packs the 5×5 rectangle uniquely, essentially by showing that large rectangles can always be packed with fewer than 6 unused squares, and that parts of the packing can be rotated.

George Sicherman also found unique solutions of packing polyominoes in cylinders:

Pentominoes in Cylinders
n \ m345678
3 none none none none none 4
4 none none 3 4 none 4
6
5 3 4 none 4 none none

Hexominoes in Cylinders
n \ m34567
3 none none none 3 3
4 none none 3 4 none
5 none 3 3
4 		4
5 		none
6 3 3
4 		5 none 6
7
7 none 4 none none 8

Heptominoes in Cylinders
n \ m34567
4 none none none 3 none
5 none none none 3
4

4
6 none 3 none none 4

7 3 none 4 4 none

George Sicherman also found unique solutions of packing polyominoes in triangles:

Pentominoes

Hexominoes (BZ) (BZ) (BZ)

Heptominoes

George Sicherman also found unique solutions of packing polyominoes in pyramids:

Pentominoes

Hexominoes

Heptominoes

George Sicherman also found unique solutions of packing polyominoes in diamonds:

Pentominoes

Hexominoes

Heptominoes

George Sicherman also found unique solutions of packing polyiamonds in triangles:

Pentiamonds

Hexiamonds

Heptiamonds

Octiamonds

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 9/26/09.