# 6 Dots...

geepamoogle
Posers and Puzzles 19 Jul '08 18:53
1. 19 Jul '08 18:531 edit
Suppose I mark a circle of a particular radius, and then mark out along its edge a number of equally spaced dots. Connecting these dots in one manner would result in a regular polygon.

Suppose then I were to count the number of distinct ways of connecting all the dots with one continuous set of straight lines, ending with the dot I started with, with rotations and mirror images being considered identical.

With 3 dots, I have one way of connecting the dots (triangle)

With 4 dots, I have two ways of connecting the dots (square or hourglass)

With 5 dots, I have 4 ways of connecting the dots (pentagon, star, "fish", or a "wave"-type pattern)

How many distinct ways can you connect 6 dots?

Remember, the dots are evenly spaced about a normal circle, and I have to connect the dots without "lifting the pen" so to speak, so patterns like a Star of David won't work, as that requires 2 sets of lines.
2. 20 Jul '08 08:35
I think you are trying to say that from one dot you must go directly to the next, in a straight line. Otherwise, the star of david would be possible, all vertices have an even degree.
3. TheMaster37
Kupikupopo!
20 Jul '08 14:55
My guess is 8, for now particular reason ðŸ™‚
4. 20 Jul '08 20:20
Originally posted by Dejection
I think you are trying to say that from one dot you must go directly to the next, in a straight line. Otherwise, the star of david would be possible, all vertices have an even degree.
Thanks. You are indeed correct, but I couldn't think of the best way to put it at the time.
5. eldragonfly
leperchaun messiah
21 Jul '08 22:28
Just doodling around i got 8.
6. smw6869
Granny
21 Jul '08 22:46
Originally posted by eldragonfly
Just doodling around i got 8.
8.000000001 to be precise!

GRANNY.
7. 23 Jul '08 21:32
I will admit to getting this one from a book, and I missed one when I tried to find them all.

There are at least 9 distinct ways to connect the 6 dots, however, not including mirrors and rotations.
8. 24 Jul '08 08:071 edit
Apologies, still working on it.
9. 24 Jul '08 21:29
If someone wishes to submit a description of the discovered methods thus far, it might help in the search for some others.

Describing the shape can be difficult, but perhaps numbering the dots clockwise from the starting dot could help.

For instance, a regular hexagon would run from 1 to 2 to 3 to 4 to 5 to 6 to 1
10. Scheel
<blank>
31 Jul '08 22:02
Originally posted by geepamoogle
If someone wishes to submit a description of the discovered methods thus far, it might help in the search for some others.

Describing the shape can be difficult, but perhaps numbering the dots clockwise from the starting dot could help.

For instance, a regular hexagon would run from [b]1 to 2 to 3 to 4 to 5 to 6 to 1
[/b]
Or a bit more dense notation (123456).
This is also what is called a cycle, there is a mathematics of cycles and I believe that it's connection with Group Theory would give an answer.

Untill I actually find that book I have no idea if it helps or what the answer is.

A very crued ansver is There is 6! permutations of (123456) but for each there are 6 that are equivalent up to rotation, further there are 6 mirror axis that transfer one immage in to an other, and finaly direction does not matter.
This would give 6! / 6 * 6*2 * 2 = 5
Since some rotations give the same as a mirroring we are obvious double counting something. So only thing gained from this is that there are more than 5 ways to do it (And you can all easyli construct 6 examples to prove that), so not much gained.