Problem of the Month (August 2006)

In September 2004, we investigated the problem of finding polypolyforms, shapes that could be tiled individually by two given polyforms. In November 2004, we generalized this to include finding shapes that could be tiled by a given polyform in multiple ways. This month we investigate the same problems for polylines, the one dimensional analogs of polyominoes.

A do n-line is a connected union of n unit line segments that intersect only at their endpoints, and form angles that are multiples of do. For two given polylines, what is the smallest figure that can be tiled individually by them? For a given subset of n-lines, what is the smallest figure that can be tiled by those n-lines but no other? And for a given polyline, what is the smallest figure that can be tiled by it in multiple ways? For all of these questions, the answer may depend on whether we allow crossings or not.

ANSWERS

George Sicherman submitted many answers and improvements.

Here are the best known compatibilities:

Compatibility of 90o Bilines
crossing:

no crossing:

Compatibility of 90o Bilines and Trilines
crossing:

no crossing:

(George Sicherman)

crossing:

no crossing:

(George Sicherman)

crossing:

no crossing:
?

Compatibility of 90o Trilines
crossing:

no crossing:

 crossing:

no crossing:

(George Sicherman)

crossing:

no crossing:

(George Sicherman)


(George Sicherman)

Compatibility of 90o Bilines and Quadlines
1 2 / 4 4 2 / 4 1 2 / 4 2 2 2 2 1 1 / ∞ 2 / 4 ∞ 4 / ∞ 2 / ∞
4 / ∞ 2 / 4 2 / 4 2 / 4 2 2 1 1 1 1 2 1 2 1 1 1
George Sicherman

Compatibility of 90o Trilines and Quadlines
3 3 3 / 12 6 / ∞ 6 / ∞ 6 / ∞ 6 / ∞ 12 / ∞ 18 / ∞ 6 9 9 / ∞ 6 ∞ 12 / ∞ 6 / ∞
3 / 6 3 3 3 3 3 3 3 3 3 3 3 / 12 3 / 12 3 / ∞ 3 / 6 3
6 / ∞ 6 3 3 / 6 6 / ∞ 6 3 6 6 3 3 3 3 ∞ 6 3
12 / ∞ 3 / ∞ 6 6 9 / ∞ 3 3 6 6 / 12 6 6 / ∞ ∞ 3 3 3 3
18 / ∞ 6 / 24 6 3 6 / ∞ 6 3 3 / 6 3 3 9 / 18 3 3 3 3 3
George Sicherman

Here are the pictures of the largest compatibilities:

Compatibility of 90o Quadlines
∞ 2 / 4 4 6 / ? 2 / 4 4 / ∞ 8 / ∞ 4 / ∞ 16 / ∞ 8 / ∞ 4 4 / ∞ 8 / ? ∞ 32 / ? 8 / ∞
2 2 2 4 2 4 2 / 4 2 / 4 4 4 4 / ∞ 2 4 / ∞ 4 / ? 4 / ?
2 2 4 4 2 / 4 4 4 2 / 4 4 4 2 ∞ 8 4 / 12
2 / 4 2 / 4 2 4 4 4 2 / 4 4 2 / ? 2 ∞ 2 / 4 2 / 4
2 2 / 4 2 2 2 2 / ? 2 2 / ∞ 4 / ? ∞ 4 / ? 2 / ∞
2 4 4 2 / 4 4 4 8 / ? 4 2 / 4 2 / 4 2 / 4
2 2 2 2 2 ∞ 2 ∞ 2 2
2 2 4 6 4 4 8 / ? 2 4
4 / ∞ 2 2 ∞ 4 / 8 2 / ? 2 / 4 2
2 2 2 4 4 2 2
2 ∞ 2 / 4 ∞ 8 / ? 12 / ?
∞ 4 ∞ 4 2
2 4 4 2
∞ 2 2
2 2
4
George Sicherman

Here is a picture of the largest compatibility:

George Sicherman found many multiple tiling numbers of the 90o pentalines:

Zucca's Problem for 90o Trilines
I L

crossing:

no crossing:

I T

crossing:

(George Sicherman)

no crossing:

(George Sicherman)

L S

crossing:

no crossing:

L U

L T

crossing:

(George Sicherman)

no crossing:

T U

crossing:

(George Sicherman)

no crossing:

T S
U S

I U

crossing:

(George Sicherman)

no crossing:

I S

crossing:

(George Sicherman)

no crossing:

I L T
I L U

crossing:

(George Sicherman)

no crossing:

L T S

I L S

crossing:

no crossing:

I T S

crossing:
?

no crossing:

I T U

crossing:
?

no crossing:

I U S

crossing:
?

no crossing:

L T U

crossing:

no crossing:

(George Sicherman)

L U S

crossing:

(George Sicherman)

no crossing:

(George Sicherman)

T U S

I L T U

crossing:

no crossing:

L T U S

crossing:

(George Sicherman)

no crossing:

I L T S

crossing:

(George Sicherman)

no crossing:

I T U S

crossing:
?

no crossing:

I L U S

crossing:

no crossing:

I L T U S

crossing:

(George Sicherman)

no crossing:

Compatibility of 60o Bilines
crossing:

no crossing:

(George Sicherman)

crossing:

no crossing:

Compatibility of 60o Bilines and Trilines
4 / ? 4 / 8 2 4 4 / 24 2 / 8 2 4 / 8 2 4 / ? 2 / ? 2
4 / 6 2 2 / 4 2 2 2 2 2 2 2 2 4 / ?
2 2 2 2 2 2 2 2 / 4 2 / 6 2 2 / 6 4 / 18
George Sicherman

Compatibility of 60o Trilines
∞ 2 / 3 2 / 4 2 3 ∞ 3 3 / ∞ 4 / ∞ 6 / ∞ ∞ 6 / ∞
3 2 2 / 3 2 3 3 2 / 3 2 2 3 4
2 / 3 2 / 3 2 / 3 3 2 6 / 24 2 / 3 3 / ? 9 / ? 2 / 3
2 2 / 3 2 3 2 2 / 3 2 / 3 2 6 / ?
3 4 2 6 / ? 4 / ? 3 ∞ 6 / ?
2 2 / 3 2 3 3 / 6 2 / ∞ 18 / ?
2 / 3 3 2 2 2 / 3 3
2 2 / 3 4 ∞ 3 / ?
3 2 3 / 6 2 / 3
3 3 6 / ?
∞ 9 / ∞
∞
George Sicherman

George Sicherman found many multiple tiling numbers of the 60o tetralines:

Zucca's Problem for 60o Bilines
I L


(George Sicherman)

I V
L V

I L V

crossing:

no crossing:
?

Compatibility of 72o Bilines
crossing:

no crossing:

(George Sicherman)

George Sicherman found many compatibilities of the 72o bilines and trilines:

George Sicherman found many compatibilities of the 72o trilines, both with crosses and without:

George Sicherman found many multiple tiling numbers of the 72o trilines:

George Sicherman also found many compatibilities of the 51o bilines:

George Sicherman also found multiple tiling numbers of the 51o bilines and trilines:

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 5/11/07.