Problem of the Month (November 2006)

This month we consider two problems about tiling polyominoes.

1. Given two polyominoes P1 and P2 with the same area, what is the smallest area of a polyomino S so that one copy of P1 and S can tile the same shape as one copy of P2 and S? Most of the answers for pentominoes are small, but some of the answers are surprisingly large. Can you find the polyominoes S for the largest cases? Can you find a polyomino S that works for the unknown case? What are the largest answers for hexominoes and larger polyominoes? What about other polyforms, like polyhexes and polyiamonds?

2. Given two polyominoes P1 and P2 with the same area, and a positive integer n, what is the smallest shape S that can be tiled by P1 in exactly n ways, and can be tiled by P2 in exactly n ways? Can you find smaller solutions for the triominoes? What are best results for larger n? What about the pairs of tetrominoes? What about other polyforms, like polyhexes and polyiamonds? What if we want S that can be tiled in n ways by P1, P2, or P3?


ANSWERS

1.
The polyomino addition solutions for tetrominoes are fairly trivial:

Tetromino Solutions
1122
111
11
1

The solutions for pentominoes are more interesting. There are several cases that require area 5 or more, one really huge solution, and one unsolved case.

Pentomino Solutions
25112242555
1111111111
1111211123
11121117
1121111
11111?
11115
4115
111
11
5
5
7 23
(Corey Plover)

In 2012, George Sicherman studied the problem for triples of pentominoes, finding these results:

The largest solutions for hexominoes are below.

Large Hexomino Solutions
6
7
8
9
(George Sicherman)
11
(George Sicherman)

12+
(George Sicherman)

Here are the unsolved cases for hexomino pairs.

Here are some large cases for heptomino pairs:

Corey Plover is not a big fan of flipping polyominoes over, so he was interested in the solutions with this restriction.

Non-Flip Pentomino Solutions
227111122242??555
1121111211111211
112112111111121
111111121111123
111211212112?
11121211221?
11212111211
2112111121
21111121?
1111112?
11111123
?111123
11111
1211
121
25
5

Here are the largest known solutions:


(George Sicherman)

(George Sicherman)

(George Sicherman)

Corey Plover noticed that there are smaller solutions if we allow king-connected polyominoes or disconnected polyominoes:


(Corey Plover)

(Corey Plover)

George Sicherman investigated pentahexes. Here are the largest solutions.

     

Here are the unsolved pentahex pairs.

George Sicherman also investigated polyiamonds. Here are the largest solutions for the hexiamonds.

And here are the largest solutions for heptiamonds.

Here are the unsolved heptiamond pairs.

George Sicherman also investigated polyaboloes. Here are the largest solutions for the triaboloes and tetraboloes.

Here are the unsolved tetrabolo pairs.

George Sicherman also investigated polylines. Here are the largest solutions for tetralines.

Here are the unsolved tetraline pairs.

George Sicherman also gave this tetraking example:

George Sicherman found that every pair of pentacubes is equable. Here are the largest solutions:

George Sicherman also investigated pentapents. Here are his solutions:

Here are the unsolved cases:


2. George Sicherman sent some solutions for this problem as well.

Here are the smallest known solutions for a domino and triomino:

Domino and Triomino Solutions

1

2

3
(GS)

4

5

6
(GS)

8

9

1

2

3
(GS)

4

5
(GS)

6
(GS)

8
(GS)

12
(GS)

16
(GS)

Here are the smallest known solutions for triominoes:

Triomino Solutions

1

2

3
(GS)

4

5
(GS)

6
(GS)

7
(GS)

8
(GS)

9
(GS)

10
(GS)

11
(GS)

12
(GS)

13
(GS)

16

18

20

Here are the smallest known solutions for tetrominoes:

Tetromino Solutions

1

2

3
(GS)

5

1

2

3

5
(GS)

6
(GS)

7
(GS)

9
(GS)

10
(GS)

1

2


1

2

4
(GS)

1

2

3
(GS)

4
(GS)


1

2

4
(GS)

6
(GS)

8
(GS)

Here are the smallest known solutions for tetriamonds:

Tetriamond Solutions

1
(GS)

2
(GS)

3
(GS)

5
(GS)

Here are the smallest known solutions for pentiamonds:

Pentiamond Solutions

1
(GS)

2
(GS)

1
(GS)

2
(GS)

1
(GS)

1
(GS)

2
(GS)

3
(GS)

4
(GS)

5
(GS)

6
(GS)

1
(GS)

2
(GS)

1
(GS)

Here are the smallest known solutions for dihexes and trihexes:

Dihex and Trihex Solutions

1
(GS)

2
(GS)

4
(GS)

1
(GS)

1
(GS)

2
(GS)

3
(GS)

4
(GS)

Here are the smallest known solutions for trihexes:

Trihex Solutions

1
(GS)

2
(GS)

1
(GS)

2
(GS)

3
(GS)

1
(GS)

2
(GS)

3
(GS)

4
(GS)

5
(GS)

6
(GS)

7
(GS)

8
(GS)

9
(GS)

10
(GS)


If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 4/19/09.