stassen@spp2.UUCP (Chris Stassen) (02/26/86)
All good net.puzzlers know that a good tic-tac-toe player will always either win or tie a game regardless of whether or not he makes the first move. Two perfect tic-tac-toe players will always end the game in a draw. Suppose we keep the same rules of playing (3x3 board, alternating turns, etc.), but change the requirements for winning. The winner of the game is the player who forces his OPPONENT to occupy three squares in a row. Is there any strategy which will always permit a player to win? If so, which one (first or second)? Or, will two perfect "toe-tac-tic" players always end the game in a draw? -- Chris PS - My thanks to math whiz Alan Murray, who co-invented this puzzle with me. (It may have been done before, but I haven't heard about it).
kwh@bentley.UUCP (KW Heuer) (02/28/86)
In article <873@spp2.UUCP> spp2!stassen (Chris Stassen) writes: >Suppose we keep the same rules of playing (3x3 board, alternating >turns, etc.), but change the requirements for winning. The winner of >the game is the player who forces his OPPONENT to occupy three squares >in a row. It is a draw under rational play. The first player can start in the center and then mirror his opponent. I don't have a quick proof for the second player, but clearly he has the advantage of having one less mark to place. The interesting thing about this game is that the optimal first move is the center, which at first glance might seem like a good point to avoid. In fact, I believe the first player loses if he doesn't play there, or if he doesn't mirror his opponent's next move. This game was analyzed in _Mathematics Magazine_ ca. 1975-1980 (sorry I can't pinpoint it; it's also possible it was in _The American Mathematical Monthly_ instead). Now, a related question. A friend (since deceased) once told me that on a 4x4 board, with the winner being the first to achieve a line of three of his own marks *and one of his opponents*, the game is a second-player win. Does anybody have further knowledge of this? (I realize the game is slightly ill-defined -- the game x22 o11 x23 o31 x44 o41 x43 o24 x21 seems to win for both players) Karl W. Z. Heuer (ihnp4!bentley!kwh), The Walking Lint
lee@ukma.UUCP (Carl Lee) (03/01/86)
> All good net.puzzlers know that a good tic-tac-toe player will >always either win or tie a game regardless of whether or not he makes >the first move. Two perfect tic-tac-toe players will always end the >game in a draw. > > Suppose we keep the same rules of playing (3x3 board, alternating >turns, etc.), but change the requirements for winning. The winner of >the game is the player who forces his OPPONENT to occupy three squares >in a row. > > Is there any strategy which will always permit a player to win? >If so, which one (first or second)? Or, will two perfect "toe-tac-tic" >players always end the game in a draw? > > -- Chris > >PS - My thanks to math whiz Alan Murray, who co-invented this puzzle with >me. (It may have been done before, but I haven't heard about it). Martin Gardner writes in chapter four of his book The Scientific American Book of Mathematical Puzzles and Diversions: "Many delightful versions of ticktacktoe do not, however, make use of moving counters. For example: toetacktick (a name supplied by reader Mike Shodell, of Great Neck, New York). This is played like the usual game except that the first player to get three in a row loses. The second player has a decided advantage. The first player can force a draw only if he plays first in the center. Thereafter, by playing symmetrically opposite the second player, he can insure the draw." "In recent years several three-dimensional ticktacktoe games have been marketed. They are played on cubical boards, a win being along any orthogonal or diagonal row as well as on the four main diagonals of the cube. On a 3 x 3 x 3 cube the first player has an easy win. Curiously, the game can never end in a draw because the first player has fourteen plays and it is impossible to make all fourteen of them without scoring." So, what about playing 3 x 3 x 3 tic-tac-toe, where the first to get three in a row loses, since there can be no tie? Carl W. Lee Department of Mathematics University of Kentucky Lexington, KY 40506 cbosgd!ukma!lee lee@ukma.bitnet lee@uky.csnet
stassen@spp2.UUCP (Chris Stassen) (03/08/86)
The game of "toe-tac-tic" is a draw for two perfect players, as
the first player chooses the center and then mirrors the second player's
moves. (The reason that I posted it is that the solution is a lot like
that of the penny puzzle). If the first player chooses any square other
than the center, then the game is a win for the second player.
And, yes, Marvin Gardner had thought of it first, so it isn't
unique work (even if it was original). Thanks to:
Kim Althoff (kima@pesnta)... For showing 2nd player win in most cases
Joe Miller (jmil@homxb).... For correct solution and pointer to Dewey 793 *
Carl W. Lee (lee@ukma)...... For pointer to Gardner
Karl Heuer (kwh@bentley)... For correct solution
* Puzzle books can be found near 793 in the Dewey Decimal System.
-- Chris