(a) One day Martha said, "I have been alive during
all or part of five decades." Rounded to the near
est year, what is the youngest she could have
been?
(b) Of all the books at a certain library, if you se
lect one at random, then there is a 90% chance
that it has illustrations. Of all the illustrations in
all the books, if you select one at random, then
there is a 90% chance that it is in color. If the
library has 10,000 books, then what is the min
imum number of books that must contain col
ored illustrations?
(c) At least how many times must you flip a fair
coin before there is at least a 50% probability
that you will get at least three heads?
(d) We have two polyhedra (i.e., solids with polyg
onal faces), all of whose edges have length 1:
a pyramid with a square base, and a tetrahe-
2.2 Strategies for Getting Started
2.2 STRATEGIES FOR GETTING STARTED 25
dron (a tetrahedron is composed of four triangu
lar faces). Suppose we glue the two polyhedra
together along a triangular face (so that the at
tached faces exactly overlap). How many faces
does the new solid have?
(e) The picture shows a square cut into two congru
ent polygons and another square cut into four
congruent polygons.
For which positive integers n can a square be
cut into n congruent polygons?
The psychological ideas presented above may seem too vague. Perhaps you ask, "How
can I learn to work hard on problems if! can't even get started?" You have already seen
four very practical strategies that address this: the penultimate step and get your
hands dirty strategies in Example 1.2 .1, and the wishful thinking and make it easier
strategies in Example 2.1.1. There is more to say about these and other ideas that help
you begin a problem solving investigation.
As we said earlier, there are two parts to any successful solution: the investigation
and the argument. Commonly, the investigation is obscured by the polished formal
solution argument. But almost always, the investigation is the heart of the solution.
Investigations are often tort uous, full of wrong turns and silly misconceptions. Once
the problem is solved, it is easy to look over your prolonged investigation and wonder
why it took you so long to see the light. But that is the nature of problem solving for
almost everyone: you don't get rewarded with the flash of insight until you have paid
your dues by prolonged, sometimes fruitless toil. Therefore,
Anything that stimulates investigation is good.
Here are some specific suggestions.
The First Step: Orientation
A few things need to be done at the beginning of every problem.
- Read the problem carefully. Pay attention to details such as positive vs. nega
tive, finite vs. infinite, etc.
