Twenty teams entered a "round robin" tournament (meaning: each team
played
each other team once). Each game resulted in a win for one of the teams,
and at the end of the tournament each team's score was the number of
games
won (that is, teams received one point for each win and zero points for
each loss). Each score turned out to be a perfect square, and more teams
scored 16 than scored 9. How many games went against form, in the sense
that the winner turned out to have a lower score than the loser?

(This is Macalester College Problem of the Week #886.)

There will be a $1 prize for the best correct solution submitted by a
midshipman.

A solution is "correct" if it answers the question correctly and
explains
the answer; a solution is "best" if it includes the clearest correct
explanation.  Solutions are due by noon on Thursday, April 1, 1999.
Submit solutions to Prof. Hanna at mathprob@nadn.navy.mil, or via the
mailbox in Chauvenet 301.

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No correct solutions came in for problem #90 (the random triangle.)  The
probability that three points chosen at random on a circle determine an
obtuse triangle is 3/4.