Bill Taylor <w.taylor@[EMAIL PROTECTED]
> writes:
> Well, it's nice to see the silly thread about what is or isn't
> a generalization of something else, is over.
You wish. :-)
> But it leaves an interesting question which hasnt been decided.
>
> Is it possible to construct a MindNinja-type game pattern,
> i.e. a fixed pattern to be decided in advance of playing,
> that both players must try to achieve, to be played on
> some Hex or Quad board, that STILL has the property
> of win/loss complementarity? (As I call it, YMMV)
> That is - that if one player makes this pattern,
> it necessarily excludes the other making it, and that one
> or other is certain to be made once the board is filled up.
>
> So; a one-piece-put-per-move game, on a predetermined
> board, with a predetermined pattern to make, which is
> the same for both players, (i.e. both colours).
>
> Equi-tasking and win-loss complementary.
> The sort of game we purists love best!
This is easy to make, albeit not very interesting: The pattern to make
is to fill the central hex of the board with your colour. Obviously,
this is complementary and will be achieved on a filled board. But it
is also a completely trivial first-player win.
This pattern includes a specific place on the board, which you might
not like. But a finite, fixed pattern that does not include
localisation constraints can not be complementary on a sufficiently
large board: Assume you have a board that includes the pattern for one
colour. Now make a copy of the board but in reverse colours and embed
both these boards in a larger board and fill out the remaining hexes
randomly. This board will now have the pattern for both sides.
So, to ensure the unique-winner property, you either have to specify
the board _and_ the pattern, or you have to include position
constraints on the pattern (such as relative to the center or edge of
the board).
Another kind of symmetry can be made in connection games: Instead of
having two players with differently coloured pieces, you have both
playing the same colour and the winner is the one that completes a
specified pattern. If the pattern can be embedded in the board, there
will be a unique winner. Proof:
1. The empty board does not contain the pattern, but the full board
will.
2. If a partially filled board contains the pattern, adding more
pieces will not change this.
3. Since pieces are never removed, the board will eventually
transition from not having the pattern to having the pattern.
An example game could be: On a hexagonal board with a hex grid, the
first to complete a connection between the central hex and the edge is
the winner.
These games are a bit like Nim, where the winner is the one to take
(or not take, depending on which variant you play) the last piece.
Since the games are deterministic and finite, they will be sure wins
for either the first or the second player, so you would probably want
to use the pie rule to make them more interesting.
Torben
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