Re: Helix, Ring, and Bracelet

On Feb 6, 7:58 am, torb...@[EMAIL PROTECTED]
 (Torben =C6gidius Mogensen)
wrote:
>
> There are also closed paths that do not encircle any axis.
>

Well, obviously.  Nitpicking aside, these are the four types.

>
> This could work (if you restrict to the above four kinds of closed
> paths).  Have you proven that it has the desired property?
>

An informal proof can be obtained by listing all possible combinations
for Black, and the corresponding possible combinations for White.

>
> In any case, the winning criteria are getting a tad complex.
>

Agreed.  I could simplify it a tad:

There are four types of closed paths one can form on a torus:
  1. R-Helix - Right handed helix
  2. L-Helix - Left handed helix
  3. Ring - Non-helical path surrounding circular axis
  4. Bracelet - Non-helical path surrounding linear axis

Winning conditions for Black:
  1. Form a ring
  2. Form an r-helix
  3. Form an l-helix and a bracelet

Winning conditions for White:
  1. Form a bracelet
  2. Form an l-helix
  3. Form an r-helix and a ring