Bill, you were right about something. Credit where credit is due.
And Torben, sorry but by the same token you were wrong.
Bill said:
"If someone makes both a N-S loop *and* NW-SE loop,
then he has *automatically* made an E-W loop as well."
I found your explanation of this confusing, Bill, but your premise is
correct.
"Here is a 'proof'.
____________________
| : : |
|..... : \..........| Dotty has a N-S loop,
| \......: ... | and also a NW-SE loop.
| : \........ |
| : \ |
|______:_______:______| "
As Bill explained, here Dotty has created a downward sloping helix and
a NS loop. In doing so he has also created an EW loop. The EW loop
could be described as starting from the left, going North at the
intersection, encircling the board with a NS loop and arriving at the
intersection again, going east at the intersection and following the
path until it meets the West side of the board again.
Torben said:
"So I'm afraid your attempt at fixing my flawed "Torex" game has
failed. Where Torex doesn't have exclusive wins on a filled board,
your variant doesn't guarantee that there will be a winner."
Yes, it turns out that my variant does guarantee that there will be a
winner, as Bill has shown.
Timeline:
1. Torben announced Torex here in the "Hex 'generalizations'" thread
on Jan 30 at 2:53 am Pacific Time
2. I announced my as yet unnamed game here in the "Salvaging Torex"
thread on Feb 3 at 7:16 pm Pacific Time.
3. Bill announced NZ here in the thread "N.Z. (a new toroidal
connection game)" on Feb 3 at 8:46 pm Pacific Time. Bill's game, if I
understand it correctly, uses a rectangular pattern grid but is
otherwise identical to my game.
Bill commendably studies and thoroughly tests his games before
announcing them. If I did that I probably could have saved myself the
trouble of "fixing" my original game which turns out not to have been
flawed after all.
Bill, we've both been wrong at times in these discussions, including
your original claim that the game played on a hexagonal grid would not
have the property of a filled board producing exactly one winner. But
in this case you were right.
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