Inverse Tic-Tac-Toe
Chris Stassen
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).
KW Heuer
>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
Carl Lee
>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.
>
>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.
>
>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
l...@ukma.bitnet
l...@uky.csnet