24 CHAPTER 2 STRATEGIES FOR INVESTIGATING PROBLEMS
[37]. These puzzles use mostly everyday ideas, but
always require a good amount of peripheral vision to
solve. Highly recommended for warm-up work.
2.1.17 In Example 2.1.9 on page 21, we proved that
any coloring which maximizes the number of balanced
wires will be integrated. Is this result "sharp;" i.e.,
must a coloring maximize balanced wires in order to
be integrated?
2.1.18 The non-negative integers are divided into
three groups as follows:A = {0,3.6,8.9 .... }. B = {I,4, 7, 11, 14, ... },C = {2. 5, 10, 13, ... }.Explain.
2.1.19 You are locked in a 50 x 50 x 50-foot room
which sits on 100-foot stilts. There is an open window
at the comer of the room, near the floor, with a strong
hook cemented into the floor by the window. So if you
had a Ioo-foot rope, you could tie one end to the hook,
and climb down the rope to freedom. (The stilts are
not accessible from the window.) There are two 50-
foot lengths of rope, each cemented into the ceiling,
about 1 foot apart, near the center of the ceiling. You
are a strong, agile rope climber, good at tying knots,
and you have a sharp knife. You have no other tools
(not even clothes). The rope is strong enough to hold
your weight, but not if it is cut lengthwise. You can
survive a fall of no more than 10 feet. How do you get
out alive?
2.1.20 Compose your own "census-taker" problem.
Invent a riddle that involves numerical information and
clues that don't seem to be clues.
2.1.21 A group of jealous professors is locked up in
a room. There is nothing else in the room but pencils
and one tiny scrap of paper per person. The profes
sors want to determine their average (mean, not me
dian) salary so that each one can gloat or grieve over
his or her personal situation compared to their peers.
However, they are secretive people, and do not want to
give away any personal salary information to anyone
else. Can they determine the average salary in such a
way that no professor can discover any fact about the
salary of anyone but herself? For example, even facts
such as "three people earn more than $40,000" or "no
one earns more than $90,000" are not allowed.
2.1.22 Bottle A contains a quart of milk and bottle Bcontains a quart of black coffee. Pour a small amount
from B into A, mix well, and then pour back from A
into B until both bottles again each contain a quart of
liquid. What is the relationship between the fraction of
coffee in A and the fraction of milk in B?
2.1.23 Indiana Jones needs to cross a flimsy rope
bridge over a mile-long gorge. It is so dark that it is im
possible to cross the bridge without a flashlight. Fur
thermore, the bridge is so weak that it can only support
the weight of two people. The party has just one flash
light, which has a weak beam, so whenever two peo
ple cross, they are constrained to walk together, at the
speed of the slower person. Indiana Jones can cross the
bridge in five minutes. His girlfriend can cross in 10
minutes. His father needs 20 minutes, and his father's
sidekick needs 25 minutes. They need to get everyone
across safely in one hour to escape the bad guys. Can
they do it?
2.1.24 You have already worked a little bit with Pas
cal's Triangle (Problem 1.3.17 on page 10). Find a
way to get Fibonacci numbers (see Problem 1.3.18 on
page 10) from Pascal's Triangle.
2.1.25 Example 1.1.2 involved the conjecture
I I lIn- +-+-+"'+--=--.
1·2 2·3 3·4 n(n+I) n+I
Experiment and then conjecture more general formu
las for sums where the denominators have products of
three terms. Then generalize further.
2.1.26 It is possible to draw figure A below without
lifting your pencil in such a way that you never draw
the same line twice. However, no matter how hard you
try, it seems impossible to draw figure B in this way.
Can you find criteria that will allow you quickly to
determine whether any given figure can or cannot be
drawn in this way?
A B2.1.27 Trick Questions. All of the following problems
seem to have obvious solutions, but the obvious solu
tion is not correct. Ponder and solve!