Alice and Bob play the following number-guessing game.  Alice writes
down
a list of ten positive integers x1, x2, ..., x10.  Bob tries to guess
Alice's numbers by asking her questions.  Bob's strategy is this: he
picks
ten positive integers a1, a2, ..., a10 and asks Alice the value of a1*x1
+ a2*x2 + ... + a10*x10.  Then Bob chooses another list of ten positive
integers b1, b2, ..., b10 and asks Alice the value of b1*x1 + b2*x2 +
...
+ b10*x10.  Play continues this way until Bob is able to determine
Alice's
numbers.

How many rounds will Bob need before he's sure of Alice's numbers?

(From Konhauser, Velleman, and Wagon, "Which Way did the Bicycle Go?") 

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 Wednesday, February 24, 1999.
Submit solutions to Prof. Hanna at mathprob@nadn.navy.mil, or via the
mailbox in Chauvenet 301.

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Two midshipmen, 3/c Friedman and 4/c Privette, submitted correct advice
for problem #87, predicting whether the bottom of a coin was heads or
tails, but neither quoted the correct odds.  So I keep my dollar this
week.  In the absence of further information, bet "tails," since 32 of
the
possible 60 coin faces are tails and only 28 are heads; your odds are
32/60 or 8/15.  If you see the top of the coin, bet the bottom is the
same, since 40 of the 60 possible coin faces have the same symbol on the
other side and only 20 have the opposite; your odds are 40/60 or 2/3.