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=45
length 15
Found by Roger Phillips

s=2, d=30
length 20
Found by Susan Hoover

s=2, d=22.5
length 32
Found 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?


ANSWERS

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

s=1/√2, d=45
length 1

s=1, d=45
length 4

s=2-1/√2, d=45
length 7

s=1+1/√2, d=45
length 10

s=2, d=45
length 15
Found by Roger Phillips

s=3-1/√2, d=45
length 17

s=1+√2, d=45
length 19

s=4-√2, d=45
length 21

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

s=√3/2, d=30
length 2

s=1, d=30
length 4

s=3-√3, d=30
length 7

s=3/2, d=30
length 10

s=(5-√3)/2, d=30
length 11

s=7/2-√3, d=30
length 13

s=1+√3/2, d=30
length 14

s=(9-3√3)/2, d=30
length 15

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

s=.708, d=22.5
length 1

s=.924, d=22.5
length 2

s=1, d=22.5
length 6

s=1.077, d=22.5
length 7

s=1.229, d=22.5
length 9

s=1.323, d=22.5
length 10

s=1.542, d=22.5
length 12

s=1.690, d=22.5
length 16

s=1.914, d=22.5
length 18

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

s=.810, d=18
length 2

s=.952, d=18
length 3

s=1, d=18
length 6

s=1.049, d=18
length 9

s=1.147, d=18
length 11

s=1.289, d=18
length 12

s=1.310, d=18
length 13


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

s=1, d=45
length 4

s=1+1/√2, d=45
length 8

s=2, d=45
length 12

s=3-1/√2, d=45
length 16

s=4-√2, d=45
length 18

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

s=1, d=30
length 5

s=3/2, d=30
length 8

s=5/2-√3/2, d=30
length 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 39
Found by Dave Langers

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

s=1, d=30
length 2

s=2, d=30
length 9
Found by Serhiy Grabarchuk

s=3, d=30
length 21
Found by Dave Langers

Here are the longest 30o snakes inside some circles:

r=1, d=30
length 12
Found by Peter Grabarchuk

r=3/2, d=30
length 33
Found 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=30
length 6

d=30
length 11

d=30
length 14

d=30
length 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
Vadim
Trofimov
34
Vadim
Trofimov
33
Leonid
Shishlo
4 33
Andrea
Concaro
43
Vadim
Trofimov
44
Vadim
Trofimov
59
Vadim
Trofimov
50
Specht,
Viertel, and
Wohlgemuth
77
Moritz
Franckenstein
70
Hermann
Jurksch
5 57
Vadim
Trofimov
81
Vadim
Trofimov
77
Hermann
Jurksch
111
Sigg and
Pfoertner
91
Mark
Beyleveld
146
Specht,
Viertel, and
Wohlgemuth
121
Hermann
Jurksch
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
Vadim
Trofimov
7 125
Jurksch and
Pfoertner
176
Hugo
Pfoertner
179
Markus
Sigg
269
Sigg and
Pfoertner
221
Jurksch and
Pfoertner
369
Sigg and
Pfoertner
288
Hermann
Jurksch
8 169
Markus
Sigg
247
Hermann
Jurksch
246
Sigg and
Pfoertner
371
Sigg and
Pfoertner
318
Pfoertner and
Jurksch
9 230
Hugo
Pfoertner
325
Sigg and
Pfoertner
328
Hermann
Jurksch
509
Sigg and
Pfoertner
10 317
Hermann
Jurksch
419
Hermann
Jurksch
444
Pfoertner and
Jurksch
11 393
Hugo
Pfoertner
533
Hermann
Jurksch
12 478
Pfoertner and
Jurksch
13 559
Hermann
Jurksch
14 681
Pfoertner and
Jurksch

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 9/1/14.