I wrote a quick c++ program to work on this problem.
(source: http://www.mathpuzzle.com/snake.zip )
It takes a short while to run, then outputs:
029663307094949024b8
03863a3a319966330949
050966330086b6b63169
25a863a3a36169900386
(0 means a movement at 0 degrees, 1 at 30 degrees, up to b at 330 degrees)
Two of these are just duplicates (I hadn't made too much of an attempt to avoid this). So, barring any bugs in my program (and I tested most of it pretty carefully), there're only two 30-angle 20-snakes, and none of greater length exist.
Now for the 3x3 ones. ;)
- Jon
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Here's a 30 degree 18-length snake. I'd hate to say how many hours I toyed
around with this puzzle, and the 45 degree version. I got both the 15's on
the 45 degree, including the symmetric, but couldn't do any better.
On the 18 length at 30 degrees, the upper edge is pretty danged close (box
height 2, internal height 1.964), so if one actually used matches, it'd hit.
I always keep a box of theoretically infinitely thin matches around, though,
for lighting imaginary fires, so it was no sweat. It almost looked like a 19
if the upper internal end could fold back to the right at 30 degrees, but I
checked the intersect and it hits. Too bad.
It was fun.
Tom Jolly
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Ed Pegg Jr wrote on 8 Apr 2003:
>
> Serhiy Grabarchuk sent me a problem involving matchstick snakes [...]
Did you know that the UnaBomber enjoyed playing with matches?
Below is one of his problems.
A Match Stick Problem, Mathematics Magazine, Jan-Feb 1971 p.41
http://links.jstor.org/sici?sici=0025-570X(197101)44:1<41:PAS>
787. Proposed by T. J. Kaczynski, Lombard, Illinois
Suppose we have a supply of matches of unit length. Let there be given a
square sheet of cardboard, n units on a side. Let the sheet be divided
by lines into n^2 little squares. The problem is to place matches on the
cardboard in such a way that: a) each match covers a side of one of the
little squares, and b) each of the little squares has exactly two of its
sides covered by matches. (Matches are not allowed to be placed on the edge
of the cardboard.) For what values of n does the problem have a solution?
Solutions:
Richard A. Gibbs: invokes Pick's theorem
Richard L. Breisch: solves m by n generalization
Math. Mag v.44 #5 Nov-Dec 1971 p.294
http://links.jstor.org/sici?sici=0025-570X(197111)44:5<294:PAS>
Thomas Wray: solves n-dim generalization (match -> n-1 dim cube)
Math. Mag. v.45 #2 Mar-Apr 1972 p.110
http://links.jstor.org/sici?sici=0025-570X(197203)45:2<110:PAS>
-Bill Dubuque
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The 64 length 19 snakes. A few are extendible to length 20.
114477A31A5A5901496, 114477A31A703A76307, 114477A3A3A703A3470, 114477A3A30707075A1, 114477A307A3A3A3470, 114477A30703A7075A1, 114477A30A5A5A13470, 114477AB2494941BA72, 114477AB272B4727941, 114477AB274B4B4BA72, 114477AB4B272727941, 114477BB4B472BA764B, 11447708B418181461A, 1146277A318A3A3A753, 1162477AB164B4B479B, 1347B692B69114477A3, 1349692B69114477A31, 1349692B69114477A30, 13A7734941196B29691, 13A7734941196B29692, 13A774411892964B418, 13A77441196B2969430, 13A77441199494169A1, 13A7744181A5A5A1350, 1448727AA25092529B4, 14581A5A5511AA7735A, 1461AA7743A3A3A7074, 149470707427AA11449, 1494946BA5A11447A3A, 1494949427AA1144961, 1496169427AA1144961, 14969114477A3192969, 14969114477A31A5A59, 1496911447701969416, 149727272B4BA774418, 14974B4B42AA77441A5, 14974B4B42AA7744027, 14974B4B4727AA11449, 14974B4B40A77441A5A, 15692B69114477A30B4, 15692B69114470A5694, 1585038511A5AB774B2, 161467A3A3A3A707494, 161A774411970707427, 1625A5A83511A807722, 1663A3A05377A081166, 168303850377A3A9116, 168305830377AA1153A, 168305058511AA7735A, 16843A3A05377AA1166, 1684411AA56B63181A5, 1684411AA636B6B8350, 1684411AA636B836B92, 169014961479411AA76, 21AA774B14949521A58, 21AA774B4B472B4BA72, 2478B4181884411AA72, 25A527AA11449A72529, 2727A5A03A3A7744118, 277AA1149614974B418, 27A5A0383277AA11538, 27AA11448B638B6A125, 27AA114947070742058, 2B477AA115830585AB2