October 21, 2000(at Williams College)
1. Suppose you start at the equator and travel continuously
northwest. How long
is your path? Roughly, what does your path look like? Assume
that the earth is a
round ball of radius 4000 miles.
2. You are told that there are two tables in a room. One
of the tables is empty,
the other has a large number of pennies on it, exactly 31 of which
are heads. You
are required to reposition the pennies using both tables so that
each table has
exactly the same number of heads showing. You must have all of
the pennies on
the tables.
You are blindfolded and cannot see the pennies, nor can you
tell be feeling
whether they are heads or tails. How should you proceed?
3. You have to get gas at one of the four stations on the
way to work this
morning. Your strategy is to observe the prices at the first
k stations and then
take the next station cheaper than all of them (or the final station
if necessary).
What value of k gives you the lowest expected (average) price?
By how much?
Assume that the prices are p, p+1, p+2, and p+3 (not necessarily in that order).
4. Consider an exponential tower of 2000 5s:
What is the remainder after dividing by 7?
5. a) What is the expected (average) length of the shorter
piece of a meter stick
with a random cut?
b) What is the expected length of the shortest of 3 pieces from
2 random cuts?
c) What is the expected length of the shortest of 4 pieces from
3 random cuts?
d) What is the expected length of the shortest of n pieces from
n-1 random cuts?
6. Prove the isoperimetric theorem, that the round circle
encloses the most area
for a given perimeter or equivalently has the least perimeter
for a given area. Or
for partial credit, show that the best quadrilateral is a square,
or that the best
parallelogram is a square, or that the best rectangle is a square.