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

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 30^o 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 22.5^o 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 18^o 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 45^o 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 30^o 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 45^o snakes inside some small rectangles:

	0	1/√2	1	2-1/√2	1+1/√2	2√2-1
1/√2
1
2-1/√2
1+1/√2
2

length 39 Found by Dave Langers

Here are the longest 30^o 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 30^o 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 30^o 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 30^o 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 \ N 5 7 8 9 10 11 12 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 Specht, Viertel, and Wohlgemuth 5 57 Vadim Trofimov 81 Vadim 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 Vadim 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.