Re: Helix, Ring, and Bracelet

"markste...@[EMAIL PROTECTED]
" <markste...@[EMAIL PROTECTED]
> wrote:

> No, Bill.  The problem that Torben identified in my earlier attempt
> applies to your game as well.  I'll translate it for you in a way that
> applies specifically to your game.

No Mark, it doesn't, as I shall explain in a moment.  It wasn't clear
which game Torben's remark was intended to apply to, but in
any event it doesn't apply to mine, nor to your "Torus"; though
I note you have wisely included redundancy in the ruleset,
to help out those who can't keep up with the logic.

> In your game N wins if he forms an N-S loop, and Z wins if he forms an
> E-W loop.  In the case that neither player achieves his loop goal and
> instead there are two helices, N wins if there is a negative gradient
> and Z wins if there is a positive gradient.
>
> The problem is that if Z makes an N-S loop *and* a helix with a
> negative gradient, he hasn't achieved either of his goals but he
> prevents N from achieving either of his.

Alas, no.  If someone makes both a N-S loop *and* NW-SE loop,
then he has *automatically* made an E-W loop as well.  In fact,
he "owns" the global board, in that his opponent will not now
be able to make a loop of *any* kind.  I didn't mention this
earlier because I thought it would have been obvious.
Sorry to be taking too much on trust.

Here is a "proof".
____________________
|            :              :           |
|.....       :               \..........|    Dotty has a N-S loop,
|     \......: ...                      |    and also a NW-SE loop.
|            :    \........             |
|            :             \            |     Blanky can have the
whole rest
|______:_______:______|    of the board, and not any loop!

Now, look at this path.  From the central intersection,
go up the vertical line, up it from the bottom, turn right
down the diagonal line, and go right around to come in
from the left again, back to the intersection.

This path has done a complete west-to-east
cir***navigation, and (a careful tracking will show!)
no N-S cir***navigation at all.

If you draw it for yourself, making the paths quite thick,
and carefully stick to the appropriate edge of each path,
(changing sides where necessary), you can even make
it look completely non-self-intersecting.

>  Now neither player can win.

So this is wrong.

> Try to keep up with the discussion.

Try to keep up with the logic, Mark!  Try to use your game
theoretic wits to the best advantage.  We know you have
them, from your many excellent games, so try your best.
And you might also try to be less touchy, egotistical, and
dismissive of others.

>*And* my game has the property that a filled board
> produces exactly one winner.

Of course we all do.  Try not to be so childishly egocentric.

> Any questions?

I hope you have no more?

-- Bill the Boardman