From: mathwft@math.canterbury.ac.nz (Bill Taylor)
Subject: 3^n TIC-TAC-TOE question.
Date: 6 Feb 1999 04:07:00 GMT
Newsgroups: rec.games.abstract,sci.math,rec.puzzles
A recent thread on tic-tac-toe brought this to mind.
Someone mentioned the enormous advantage of playing first, in the centre,
in 3x3x3x3x...x3 tic-tac-toe. And obviously so.
Someone (Martin Garner?) once observed that, oddly, starting first,
with the SAME MOVE, is also a non-losing strategy at misere tic-tac-toe,
rather unexpectedly.
In misere tic-tac-toe the first to make a line of 3 loses. Starting first,
just take the centre, then play diametrically opposite the secondmover,
and you can't lose! Cute.
But it MAY be a tie, of course. So, if firstplayer uses this strategy...
Is it always possible for secondplayer to force a tie?
=====================================================
In 2 dimensions, trivially so. I played myself a quick game of
3x3x3, (3 dimensions, or rather, 3 dementions, as someone serendipitously
miswrote in another thread just the other day). I found myself losing in
spite of all I could do. I wasn't trying hard, though. :)
One gets the feeling that in sufficiently many dement... dimensions,
a 3-line becomes inevitable.
Anyone got any ideas or refs?
-------------------------------------------------------------------------------
Bill Taylor W.Taylor@math.canterbury.ac.nz
-------------------------------------------------------------------------------
The Germans were very good at winning battles, but disastrous at making war.
The British were always exactly the reverse.
-------------------------------------------------------------------------------
==============================================================================
From: hale@mailhost.tcs.tulane.edu (Bill Hale)
Subject: Re: 3^n TIC-TAC-TOE question.
Date: 6 Feb 1999 06:34:07 GMT
Newsgroups: rec.games.abstract,sci.math,rec.puzzles
In article <79gf54$brs$1@cantuc.canterbury.ac.nz>,
mathwft@math.canterbury.ac.nz (Bill Taylor) wrote:
> A recent thread on tic-tac-toe brought this to mind.
>
> Someone mentioned the enormous advantage of playing first, in the centre,
> in 3x3x3x3x...x3 tic-tac-toe. And obviously so.
This is not an answer to your question, but it is an aside to the
importance of the center square.
There is a version of tic-tac-toe called Achi, where each player gets
four pieces and the play precedes as in standard tic-tac-toe with the
same goal. After the eight pieces have been put down and nobody has
won yet, then play continues by a player moving one of his pieces into
the vacant square, trying to get 3-in-a-row (horizontal, vertical, or
diagonal). (If a player can't move, then he loses.)
In this game of Achi, the first player always wins, but playing the
first piece to the center square by the first player is a losing move.
A game will last no more than 15 or 16 moves.
I have a freeware version of this game that runs on a Macintosh computer.
It is very difficult to win, even if you go first. It is more of a
puzzle than a strategy game.
--
Bill Hale
==============================================================================
From: hale@mailhost.tcs.tulane.edu (Bill Hale)
Subject: Re: 3^n TIC-TAC-TOE question.
Date: 6 Feb 1999 07:54:15 GMT
Newsgroups: rec.games.abstract,sci.math,rec.puzzles
In article ,
hale@mailhost.tcs.tulane.edu (Bill Hale) wrote:
> In article <79gf54$brs$1@cantuc.canterbury.ac.nz>,
> mathwft@math.canterbury.ac.nz (Bill Taylor) wrote:
>
> > A recent thread on tic-tac-toe brought this to mind.
> >
> > Someone mentioned the enormous advantage of playing first, in the centre,
> > in 3x3x3x3x...x3 tic-tac-toe. And obviously so.
>
> This is not an answer to your question, but it is an aside to the
> importance of the center square.
>
> There is a version of tic-tac-toe called Achi, where each player gets
> four pieces and the play precedes as in standard tic-tac-toe with the
> same goal. After the eight pieces have been put down and nobody has
> won yet, then play continues by a player moving one of his pieces into
> the vacant square, trying to get 3-in-a-row (horizontal, vertical, or
> diagonal). (If a player can't move, then he loses.)
I made a mistake here. A player doesn't win by getting 3 pieces aligned
diagonally. A player wins only by getting 3-in-a-row horizontally or
vertically.
>
> In this game of Achi, the first player always wins, but playing the
> first piece to the center square by the first player is a losing move.
>
> A game will last no more than 15 or 16 moves.
>
> I have a freeware version of this game that runs on a Macintosh computer.
> It is very difficult to win, even if you go first. It is more of a
> puzzle than a strategy game.
>
> --
> Bill Hale
==============================================================================
From: Felix Lee
Subject: Re: 3^n TIC-TAC-TOE question.
Date: 06 Feb 1999 08:51:09 -0800
Newsgroups: rec.games.abstract,sci.math,rec.puzzles
mathwft@math.canterbury.ac.nz (Bill Taylor) writes:
> In misere tic-tac-toe the first to make a line of 3 loses. Starting first,
> just take the centre, then play diametrically opposite the secondmover,
> and you can't lose! Cute.
>
> But it MAY be a tie, of course. So, if firstplayer uses this strategy...
>
> Is it always possible for secondplayer to force a tie?
> =====================================================
>
> In 2 dimensions, trivially so. I played myself a quick game of
> 3x3x3, (3 dimensions, or rather, 3 dementions, as someone serendipitously
> miswrote in another thread just the other day). I found myself losing in
> spite of all I could do. I wasn't trying hard, though. :)
3x3x3 is always a lose for the second player if the
first player always plays diametrically opposite.
consider a corner view of the cube centered on a corner
taken by O:
.
. .
. . .
. .
. O .
. .
. . .
. .
.
in order to have a tie, we have to fill this diagram with Xs
and Os without making 3-in-a-row. a 3-in-a-row of X means a
lose for O on the opposite side of the cube, since O is
forced to make the opposite 3-in-a-row first.
because of player 1's strategy, the six corners in this
diagram must have 3 Xs and 3 Os in a symmetric pattern.
there are two possible configurations:
(1) O (2) X
. . . .
X a X X . X
. . . .
. O . . O .
. . . .
O . O O . O
. . . .
X O
in (1), the space marked (a) will force a 3-in-a-row.
in (2), if you block the immediate 3-in-a-rows, you get:
X
O O
X O X
b .
. O .
X X
O X O
X X
O
and the space marked (b) will force a 3-in-a-row.
--