Here is my first attempt (for this year!) to construct
a toroidal and locally-hex form of connection game.
It is not really satisfactory, as we will see.
The type of board that I'm REALLY keen on(!), is this:-
| . . . .
| . . . . .
| . . . . . .
| . . . . . . .
| . . . . . . . .
| . . . . . . .
| . . . . . .
| . . . . .
Obviously this can extend to any size,
with alternately n and n+1 length edges.
This has THREE glueing connections, rather than just 2;
top to bottom, NE to SW, NW to SE. They glue while
maintaining the locally-hex connections. As shown here:
| o n m l k
| h A B C D r The original board is shown with
| i U . . . E q the rim cells marked with caps.
| j T . . . . F p
| k S . . . . . G o Surrounding is an outer rim of
| d R . . . . . . H a "ghost cells" in lowercase,
| e Q . . . . . I u such that when a real rim cell
| f P . . . . J t is played, a similar stone
| g O N M L K s automatically appears in its
| h a b c d r ghost of the same letter.
So, standard sort of direct, cross-board connections.
It looks at first as if this might be some sort of
"super-torus", but mathies will easily recognize it
as merely a different representation of a 4x12 hex-torus.
It has much nicer symmetry properties this way, though!
To win, a player must make a global loop of
the appropriate type. That is, NW/SE, NE/SW, or N/S.
That is unfortunately three types for just two players!!
(Nor does a 3-player version work, alas!)
One possible solution is to play 2 winning-loops for one
player, and 1 winning-loop for the other. This would require
having both players play cells for the less-favoured player,
until someone says: "Enough, I take the less-favoured!"
This would add an unbalanced "chicken"-like element,
as in Unlur. (NB, Joao and I play a lot of these
unbalanced variants, BTW - and they're good fun!)
To keep the players evenly balanced, it can be done,
but at the cost of some unsatisfactory direction bias.
Easiest-to-view is to make the N/S direction the biassed one.
Player 1 wins if he makes a NW/SE loop; otherwise
player 2 wins if he makes a NW/SE loop; otherwise
either player wins if he makes a N/S loop that is
"further west" than the other player's similar loop
(if he can make one).
As an example of criterion 3, here is one...
| . o x .
| . o . x . They have both made a N/S loop,
| o o . . x . but clearly o's is further west,
| . o . . x . . so he wins.
| . . o . x . . . (And X has no chance of
| . o . x . . . a more westerly one now.)
| . o . x . .
| . o . x .
Unfortunately there are more difficult cases where
a N/S loop is partly on the west side and partly on the east,
making comparison more difficult, though not impossible,
just more "artificial".)
| . o x .
| . o . x . Both have a N/S loop.
| o o . . x .
| x . . . x o o
| x x x x x o x x
| x . . x . o o
| o o . x . .
| . o . x .
The simplest criterion would be to award victory to
whoever has the most westerly path that is entirely
on the centre part of the board. In the above case, X.
OC, x has part of his GROUP on the other side, but it
contains within it a N/S path which is entirely within
the board itself - no E/W crossings. So he wins.
_ _ _
Well, there it is. No name for this game as yet.
Pity it has a slightly artificial criterion -
the "more westerly" attribute.
So, not as "natural" as NZ.
-- Bill the Board-gamer
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