# Problem of the Month (September 2004)

This month we consider the problem of finding the smallest region that can be tiled independently by two different polyforms. This sort of problem has already been investigated for polyominoes.

The problem for the 3 trihexes is trivial, since a pair of any of them will tile the same region. The problem for the 7 tetrahexes is more interesting, but still pretty easy. What are the results for the 22 pentahexes below?

The corresponding problem for tetriamonds and pentiamonds has been studied. What are the results for the 12 hexiamonds below?

The problems for diaboloes and triaboloes are pretty easy. What are the best results for the 14 tetraboloes below?

The problem for the generalized triominoes is trivial. What are the results for the 22 generalized tetrominoes?

## POLYHEXES

Mike Reid improved one of my tetrahex results.

Joe DeVincentis sent a complete solution to the tetrahex problem, and found many of the best known solutions for the pentahex problem. Jeremy Galvagni sent some pentahex solutions. George Sicherman found better solutions for several pairs of pentahexes.

Trihexes
 2 2 2

Tetrahexes
 2 2 2 2 3 ∞ 2 2 2 2 6 2 2 2 2 2 2 2 2 2 3
Mike Reid

Pentahexes
 X R L P E H F Y C Q A T K W Z V U N J S I D 3 2 2 2 3 3 3 2 3 2 2 6 2 2 2 3 2 2 2 2 2 X 2 8 2 2 3 6 2 30 2 3 60 2 6 4 24 78 3 3 15 35 R 2 2 2 3 2 2 2 2 2 2 2 4 2 2 3 2 2 2 3 L 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 P 2 2 2 2 2 2 2 6 2 2 2 2 2 2 2 3 2 E 2 3 2 3 2 2 3 2 2 2 18 3 2 4 2 2 H 2 6 3 2 2 2 2 2 2 11 3 2 2 2 10 F 6 2 3 2 2 2 2 2 6 2 2 2 2 9 Y 3 2 2 2 2 6 2 2 6 2 2 2 2 C 2 4 2 2 3 2 2 3 2 2 2 9 Q 3 2 2 2 2 4 6 3 2 2 8 A 6 2 8 3 2 6 2 2 6 3 T 2 6 2 2 3 3 3 2 ∞ K 2 2 3 3 2 2 2 8 W 2 10 2 2 3 4 5 Z 2 2 2 2 2 2 V 6 2 2 3 3 U 2 2 2 18 N 2 3 3 J 2 3 S 5
Joe DeVincentis
George Sicherman

The pictures of the largest compatibility regions for pentahexes are here, and hexahexes are here: 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10, courtesy of George Sicherman.

## POLYIAMONDS

George Sicherman sent me this link to an abstract of Margareta Lukjanska's on what she calls the compatibility problem.

Margarita Lukjanska sent me her results on hexiamonds, many of which improved on my results. She and Mike Reid both gave me the reference of "Puzzle Fun" published in August 1995 by R. M. Kurchan from Buenos Aires, which published Mike's results on the problem.

Hexiamonds
 L E P S F V H A U O X I 3 3 2 3 2 2 3 2 3 ∞ ∞ L 3 2 3 2 3 2 3 2 6 3 E 6 4 6 6 2 3 3 ∞ 3 P 2 2 2 2 2 2 6 2 S 2 3 2 18 2 3 ∞ F 2 2 2 2 6 6 V 3 2 3 3 ∞ H 3 2 6 3 A 3 ∞ ∞ U 3 ∞ O ∞
Margarita Lukjanska
George Sicherman

Here is a picture of the largest case:

Scott Reynolds sent me these results on hexiamonds pairings that are infinite.

## POLYABOLOES

Diaboloes and Triaboloes
 4 ∞ 2

 4 4 2 2 2 2

Tetraboloes
 Z D A L R C Y K O S G V J I ∞ 4 2 ∞ 4 ∞ ∞ 4 ∞ ∞ ∞ ∞ ∞ Z 2 2 2 ∞ ∞ ∞ 4 ∞ ∞ ∞ 2 ∞ D 4 4 2 12 ∞ 2 ∞ ∞ 2 4 2 A 2 4 6 ∞ 2 4 4 4 ∞ 4 L 4 ∞ 2 4 ∞ ∞ 2 ∞ 2 R 2 4 4 4 4 2 ∞ 4 C ∞ 2 ∞ ∞ 4 ∞ 4 Y 2 ∞ ∞ ∞ 4 ∞ K ∞ 4 4 16 2 O ∞ 2 ∞ 2 S 2 ∞ 2 G 2 2 V 2
George Sicherman

The 5-aboloes are here, here, and here.

## GENERALIZED POLYOMINOES

Generalized Triominoes
 2 2 2 3 2 2 2 2 2 2

Generalized Tetrominoes
 Q L T O N M Z A B D R J C E Y P S H V X F I 2 2 2 2 2 4 2 12 2 4 4 4 2 4 4 2 2 4 4 ∞ 4 Q 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 2 4 4 L 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 8 4 T 4 2 2 2 2 4 2 4 2 4 4 4 2 2 2 2 2 4 O 2 2 4 4 2 4 4 4 2 4 4 2 2 4 2 ∞ 4 N 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 M 2 2 2 2 2 2 2 2 2 2 4 2 2 4 2 Z 2 2 2 2 2 2 2 2 2 2 4 2 2 2 A 2 2 4 2 2 2 2 2 2 4 2 2 4 B 2 2 2 2 4 2 2 2 4 2 8 4 D 2 2 2 2 2 2 4 2 2 4 2 R 2 2 2 2 2 4 2 4 8 4 J 2 2 2 2 2 2 2 2 4 C 2 4 2 2 4 2 4 2 E 2 2 2 4 2 2 2 Y 2 2 2 2 2 4 P 2 2 2 2 4 S 2 2 4 2 H 2 4 2 V 2 2 X 2
George Sicherman

## POLYPENTS

George Sicherman considered tripents, and suggested I look at tetrapents. So I did. Here are the results.

Tetrapents
 4 2 4 2 2 6 3 2 6 2 2 4 6 5 6 2 4 2 2 5 6
George Sicherman

Then Scott Reynolds and George Sicherman tackled pentapents. The results can be seen at George Sicherman's site.

## POLYHEPTS

George Sicherman considered tetrahepts. Here are the results.

Tetrahepts
 7 4 4 2 2 2 4 4 6 7 6 4 6 2 4 2 4 4 2 6 7
George Sicherman

## POLYOCTS

George Sicherman also considered tetraocts. The results are here.

## POLYNONS

George Sicherman also considered trinons and tetranons. The results are here, here, and here.

## POLYDECs

George Sicherman also considered tetradecs. The results are here and here.

## ZUCCA'S PROBLEM

George Sicherman suggested Zucca's problem: given a subset of these polyforms, find the smallest shape that can be covered by polyforms in the subset, but NOT by polyforms NOT in the subset. He investigated the problem for tetrahexes and generalized triominoes. Here is his page of results. And here are the best known solutions for trihexes, diaboloes, and generalized triominoes.

Several of the generalized triomino cases are due to George Sicherman.

George Sicherman also investigated tetracubes:

George Sicherman has many pages devoted to this problem here.

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 8/6/08.