The Problem Solving Competition
Cyrus Hsia
NOVEMBER 1998
The Problem Solving Competition
THE FOUR SPHERE PROBLEM
Suppose that three identical spheres of radius R are sitting on a
flat, level table such that each touches the other two. Suppose that a
fourth, smaller sphere of radius r is sitting on the table in the gap
made by the three larger spheres in such a way that it just touches
each of the larger spheres at some height, h, above the table.
Express r and h in terms of R.
MAXIMUM NUMBER OF CODES
Jim Smithers has been asked to create 6-digit security codes (e.g.
383201) for all of the students using the science lab this semester by
choosing from the digits 0 through 9 inclusive for each of the 6
entries. Of course none of the 6-digit sequences are to be alike.
As a further precaution, it is stipulated that no two codes are to
differ in exactly one place. That is, if 123456 is a particular
student's code, 123956 cannot be used for a different student. There
are many students using the lab this year. How many such student
codes can be made?
That is, what is the maximum number of 6-digit codes that can
be made by Jim according to these restrictions?
Give justification for your answer.
(These are also posted on the bulletin board outside the Math Office.)
Have fun!
Cyrus Hsia
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The Problem Solving Competition
This math competition is open to any undergraduate. Two math
problems will be given each month. At the end
of this year, two winners from each undergraduate institution will
go to compete in Providence, Rhodes Island, US. Do UofT proud, try
the problems.
Solutions due: December 15, 1998.
Please send solution via email to Cyrus Hsia, hs...@math.utoronto.ca
OR
in my mailbox next to the Math Office in Sidney Smith Hall, 4th floor.