The HEXA-Y problem.
Bill Taylor
(a) to make a connection game on a Hex-hex board,
(b) which is win/loss complementary, (like Hex, Atoll, and Y),
(c) whose winning criterion includes some sort of Y connection,
(d) but EXcludes winning merely by crossing between opposite sides.
Various attempts have been made.
_ _ _ _ _ _ _
Cross, by Cameron Browne, was promising, but turns out to be broken.
In Cross, it is merely ILLEGAL to join two opposite sides,
and wins are executed by connecting 3 alternately-spaced sides.
Alas, this leads to "poisoned cells", with concommitant pass fights
and cold moves (Nim style). Though playable, it is not within
the spirit of regular connection games, where every move is hot,
i.e. passes are irrelevant and cannot gain.
This situation in Cross is typical:-
| x x x
| o o o o
| . . . . .
| x x x x
| o o o
One might call this the "corporals' stripes" pattern.
Note that with these four stripes, a standard Y win becomes
impossible for either player, and the centre cells will fill up
irrelevantly, and whoever first has to fill a POISONED CELL at
the east or west corner, automatically loses; (or alternatively
the game is drawn, depending on the exact rule wording).
Altogether an unsatisfactory situation.
_ _ _ _ _ _ _ _
A more recent attempt has been made by myself, Joao Neto,
and Nick Bentley; but has, as I have just discovered,
also a failing (and I expect common) situation.
Our attempt was to extend the crossing prohibition,
to ANY similar situation, with the rule wording that -
"No move shall be made that physically prohibits the construction
of a Y-group by the opponent, unless that move immediately
wins for the player".
In other words, you may not utterly block your opponent, except
by making a winning group yourself. As always, a winning group
is one that touches three alternately-spaced sides.
So neither player would be permitted to complete his second stripe,
in the above board picture. Nor could they make a simple crossing.
ALAS!!! This also fails.
| x o o
| o x x x
| x o . o o
| x x o x
| o . x
Here, neither player may play in the centre, or the bottom,
without thereby making an illegal cross; and they cannot
make a Y group without getting to play in BOTH desired cells.
_ _ _ _ _ _ _ _ _
So - the obvious thought is to allow either player to play
TWO SIMULTANEOUS cells, provided that that thereby wins instantly.
!! Is this a winner? :-)
!
ALAS & ALACKADAY......
| o o o o o
| x x x x x x
| . . . . . . . irrelevant rubbish in this row
| x x x x o o o o
| o o o o . x x x x
| x x x x o o o o
| o o o . x x x
| x x x o o o
| o o . x x
Neither player may play in any ONE or TWO (simultaneously)
of the three central lower cells. To do so would be to
make illegal corporal stripes as in the very first diagram.
:
And clearly this type of pattern can be extended (on larger boards)
to prohibitions on THREE (or more) simulataneous winning moves.
The idea of simultaneous moves, even for an immediate win, is not
a pleasant one for most game players anyway, but these examples
show that it would not suffice, even so.
_ _ _ _ _ _ _ _
These sorts of examples convince me that the dream of finding
a game to satisy all of (a) to (d) above is chimerical; the fact
is that there is a sort of "PARITY CURSE" on connection games.
For games with no pre-owned sides, such as Y, there must be
an odd number of sides to make it playable. Y can easily be
extended to boards with 5, 7 etc sides. (We have played these.)
OTOH, for games with "pre-owned sides", such as Hex and Atoll,
there must be an "evenly even" number of sides, i.e. a multiple
of 4. Hex has 4, Atoll 8; and we have played super-atoll with
12 and 16 sides, all very playable (with restricted
multi-move modifiers to speed them up for email).
This leaves boards with "oddly even" numbers of sides,
such as 6 and 10, quite without playable connection games for them!
This is THE PARITY CURSE of connection games: NO (4n+2) !
======================================= """""""""""""
- it is distantly similar to the curse of Graeco-Latin squares
and of finite Projective Geometries!
-- Baffled Bill
*** I'm NOT a misogynist, you stupid cow!
Nick Bentley
damning examples.