The Best of Creative Computing Volume 1 (published 1976)

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Puzzles and Problems for Fun (calculus puzzle, geometry puzzle)

graphic of page

ln general, the second player should attempt to
play so that for every missing marker the symmetrically
opposite marker is also missing. The center
marker must also be missing. lf the second player
succeeds in obtaining this board configuration at
the end of any turn, he has successfully taken
advantage of the first player's error and has a
winning strategy. On all subsequent turns he
should remove only those markers symmetrically
opposite those removed by his opponent. Following 
this strategy, if the first player's opening play is
3, 8, 13 the second player's play should be 18, 23.

But what should the second player do if the
opening play is 16- 18? That's part of the chalIenge
of the problem! Perhaps play 9 - 10, but that
seems to increase the first player's chances of
making a winning move next time. We do assume
the first player is smart even if he does err on his
first play. Perhaps play at random, but that seems
to decrease the second player's chance of obtaining
a winning board configuration.

The complexity of the problem is indeed increased 
by letting the first player be human. The
problem is very good because it is a mini-version of
what one often faces in much larger problems: the
solution is not trivial; although each step of a
solution can be well defined, some definitions will
reflect the problem solver's best judgment rather
than an absolute truth; once a solution is well
defined, a program can be written that plods
through many cases while another can be written
that uses reflections and rotations of the board to
reduce the number of cases. The challenge of
writing a program that plays Tac Tix with a smart
but fallible user who is given the first move
properly belongs under the title "Creative Computing." 
And those who write such a program are
likely to have done some "creative analysis" before
they finish.

A modified version of Tac Tix that looks easier
but is actually much more complex is played on a
4 x 4 board rather than a 5 x 5 board. The only
other change is that the player who removes the
last marker is the loser. ls there a winning strategy
for either player? After trying to define a winning
strategy for one of the players, one may well
become interested in writing a program that
develops its strategy by learning as it plays. By
repeating successful plays and avoiding the repetition 
of unsuccessful plays, the computer can
improve its strategy with each successive game. The
writing of such cybernetic programs will be the
subject of a future column.

Related References
Gardner, Martin; Mathematical Puzzles and Diversions; 
New York: Simon and Schuster; 1959;
Chapter 15.

Spaulding, R. E.; "Recreation: Tac Tix"; The
Mathematics Teacher; Reston, Virginia: National
Council of Teachers of Mathematics;
November 1973; pages 605-606.


Puzzles and
Problems for Fun


This puzzle is calculated to test your ability in
calculus: A watchdog is tied to the outside wall of
a round building 20 feet in diameter. If the dog's
chain is long enough to wind halfway around the
building, how large an area can the watchdog

A. G. Canne
Pittsburgh, Pa.


The Sheik of Abba Dabba Dhu wears this
medallion, on which each equilateral triangle
represents a wife in his harem.How many wives
does the sheik have?

David Lydy
Cincinnati, Ohio


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