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. --