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?


ANSWERS

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
22
2
     

Tetrahexes
22223
22226
2222
222
22
3
Mike Reid

Pentahexes
XRLPEHFYCQATKWZVUNJSI
D322233323226222322222
X28223623023602642478331535
R2223222222242232223
L222222222223222222
P22222226222222232
E23232232221832422
H26322222211322210
F62322222622229
Y3222262262222
C242232232229
Q32222463228
A6283262263
T26223332
K22332228
W21022345
Z222222
V62233
U22218
N233
J23
S5
Joe DeVincentis
George Sicherman

The pictures of the largest compatibility regions for pentahexes are here, and hexahexes are here, here and here, 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
LEPSFVHAUOX
I332322323
L3232323263
E64662333
P22222262
S2321823
F222266
V3233
H3263
A3
U3
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
               
442
22
2

Tetraboloes
ZDALRCYKOSGVJ
I4244
Z22242
D442122242
A24624444
L42422
R2444424
C244
Y24
K44162
O22
S22
G22
V2
George Sicherman

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

GENERALIZED POLYOMINOES

Generalized Triominoes
2223
222
22
2

Generalized Tetrominoes
QLTONMZABDRJCEYPSHVXF
I222224212244424422444
Q22222222222222224244
L2222222222222222484
T422224242444222224
O2244244424422424
N2222222222222224
M222222222242242
Z22222222224222
A2242222224224
B222242224284
D22222242242
R2222242484
J222222224
C24224242
E2224222
Y222224
P22224
S2242
H242
V22
X2
George Sicherman

POLYPENTS

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

Tetrapents
424226
32622
4656
242
25
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
744222
44676
4624
244
26
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.