Problem of the Month (May 2003)

This month's problem comes from my favorite math site: mathpuzzle.com. A snake is a sequence of unit line segments that are non-overlapping except that each one begins where the previous one ends. A d degree snake is a snake all of whose angles with the horizontal are multiples of d. We are interested in the longest d degree snake that will fit inside a square of side s.

The discussion at mathpuzzle.com has mostly been about the longest snakes that can fit inside a square of side 2:

 s=2, d=45length 15Found by Roger Phillips s=2, d=30length 20Found by Susan Hoover s=2, d=22.5length 32Found by Jon K McLean

What are the longest 30o snakes you can find in some small squares? How about 22.5o snakes? What about the longest snakes that fit inside rectangles? circles?

Here are the longest known 45o snakes inside small squares:

 s=1/√2, d=45length 1 s=1, d=45length 4 s=2-1/√2, d=45length 7 s=1+1/√2, d=45length 10 s=2, d=45length 15Found by Roger Phillips

 s=3-1/√2, d=45length 17 s=1+√2, d=45length 19 s=4-√2, d=45length 21

Here are the longest known 30o snakes inside small squares:

 s=√3/2, d=30length 2 s=1, d=30length 4 s=3-√3, d=30length 7 s=3/2, d=30length 10 s=(5-√3)/2, d=30length 11

 s=7/2-√3, d=30length 13 s=1+√3/2, d=30length 14 s=(9-3√3)/2, d=30length 15

Here are the longest known 22.5o snakes inside small squares:

 s=.708, d=22.5length 1 s=.924, d=22.5length 2 s=1, d=22.5length 6 s=1.077, d=22.5length 7 s=1.229, d=22.5length 9 s=1.323, d=22.5length 10

 s=1.542, d=22.5length 12 s=1.690, d=22.5length 16 s=1.914, d=22.5length 18

Here are the longest known 18o snakes inside small squares:

 s=.810, d=18length 2 s=.952, d=18length 3 s=1, d=18length 6 s=1.049, d=18length 9

 s=1.147, d=18length 11 s=1.289, d=18length 12 s=1.310, d=18length 13

Here are the longest known 45o snake loops inside small squares:

 s=1, d=45length 4 s=1+1/√2, d=45length 8 s=2, d=45length 12

 s=3-1/√2, d=45length 16 s=4-√2, d=45length 18

Here are the longest known 30o snake loops inside small squares:

 s=1, d=30length 5 s=3/2, d=30length 8 s=5/2-√3/2, d=30length 12

Here are the longest 45o snakes inside some small rectangles:

01/√212-1/√21+1/√22√2-1
1/√2
1
2-1/√2
1+1/√2
2
Here is the longest 30o snake inside a 2x3 rectangle:

 length 39Found by Dave Langers

Here are the longest 30o snakes inside some equilateral triangles:

 s=1, d=30length 2 s=2, d=30length 9Found by Serhiy Grabarchuk s=3, d=30length 21Found by Dave Langers

Here are the longest 30o snakes inside some circles:

 r=1, d=30length 12Found by Peter Grabarchuk r=3/2, d=30length 33Found by Dave Langers

Trevor Green and Jeremy Galvagni worked on an efficient snake in rectangles of width 1 and d small. The basic idea is shown below:

Serhiy Grabarchuk found the longest 30o snakes inside regular polygons:

 d=30length 6 d=30length 11 d=30length 14 d=30length 18

Serhiy Grabarchuk also found the longest 30o snake on the surface of a unit cube. The red lines are on the front faces, and the blue lines are on the back faces.

In November 2006, Al Zimmerman held a contest to find the largest possible 360/N degree snakes inside a circle of diameter D, for small N and D. Below is a table showing the best results.
Lengths of Longest Snakes
D \ N5789101112
2   7
Andrea
Concaro
8
Andrea
Concaro
11
Andrea
Concaro
9
Andrea
Concaro
11
Andrea
Concaro
12
Peter
Grabarchuk
3 17
Andrea
Concaro
20
Andrea
Concaro
22
Andrea
Concaro
27
Andrea
Concaro
23
Trofimov
34
Trofimov
33
Leonid
Shishlo
4 33
Andrea
Concaro
43
Trofimov
44
Trofimov
59
Trofimov
50
Specht,
Viertel, and
Wohlgemuth
77
Moritz
Franckenstein
70
Specht,
Viertel, and
Wohlgemuth
5 57
Trofimov
81
Trofimov
77
Specht,
Viertel, and
Wohlgemuth
111
Sigg and
Pfoertner
91
Mark
Beyleveld
146
Specht,
Viertel, and
Wohlgemuth
120
Markus
Sigg
6 88
Markus
Sigg
122
Specht,
Viertel, and
Wohlgemuth
123
Michael
van Fondern
192
Specht,
Viertel, and
Wohlgemuth
148
Sigg and
Pfoertner
241
Pfoertner,
Rosenthal,
and Sigg
198
Trofimov
7 124
Specht,
Viertel, and
Wohlgemuth
173
Markus
Sigg
179
Markus
Sigg
269
Sigg and
Pfoertner
220
Sigg and
Pfoertner
369
Sigg and
Pfoertner
279
Markus
Sigg
8 168
Moritz
Franckenstein
243
Sigg and
Pfoertner
244
Sigg and
Pfoertner
371
Sigg and
Pfoertner
311
Markus
Sigg
9 226
Sigg and
Pfoertner
325
Sigg and
Pfoertner
322
Sigg and
Pfoertner
509
Sigg and
Pfoertner
10 313
Markus
Sigg
417
Sigg and
Pfoertner
430
Markus
Sigg
11 367
Sigg and
Pfoertner
509
Sigg and
Pfoertner
12 457
Sigg and
Pfoertner
13 555
Sigg and
Pfoertner
14 649
Markus
Sigg

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 8/23/08.