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Introduction 5
c
A
B
Fig.
0.1. Triangle
ABC
with an interior point D
The
issue is more serious
than
it first appears.
Wehave
given this very prob-
lern to many people, including undergraduate and graduate students, and even
full professors in mathematics, engineering, or computer science. Fewer
than
five percent of
them
solved this problern within an hour, many of
them
required
several hours, and
we
witnessed some failures as well.
What's
interesting here
is
the
fact
that
this problern
was
selected from a
math
text
for fifth graders in
the
United States.
It's
easy to solve when placed
at
the
end of
the
appropriate
chapter!
But
outside of this context, it can be quite difticult.
At this stage we'd like to encourage you to
put
the
book aside and
try
to
solve this problem. To reduce your temptation to simply "keep reading" without
thinking,
we
placed the solution to this problern in
the
final section of this text.
If
you
can't
solve it within an hour, this book
is
probably for you!l
Let's consider another puzzle. There
is
a well, open
at
the
top, with a
diameter of three meters.
We
throw two sticks with lengths of four and five
meters, respectively, into the well. Let's assume
that
these sticks are positioned
as
shown in figure 0.2, i.e., they stay in the same plane and they "cross each
other" as shown.
The
problern
is
really two-dimensional, as
the
well can be
replaced by a reetangle
that
is
open
at
the
top.
Now, the challenge
is
to determine the distance h from
the
bottom
of
the
well to
the
point where
the
two sticks intersect.
The
position and
the
lengths
of the sticks guarantee a unique solution, and it seems
that
the
problern
is
easy. lndeed,
it
is
easy; it takes less
than
aminute
to solve and you shouldn't
be surprised to learn
that
the
problern
is
again taken from a primary school
curriculum. However,
the
associated "chapter contents" for this problern are
once again missing, and once more
we
encourage you,
the
reader, to solve it.
If
you solve it within an hour (!), you'll belong to
the
elite one percent of
the
people
we
tested who managed to get
the
right answer within
that
time.
What's
more,
everyone
we
tested had
at
least an undergraduate degree either in mathematics,
1
Note, however,
that
we
already gave you a valuable piece of inforrnation: the problern
is
elernentary, so there's no need
to
use any "sophisticated" tools.
Without
this hint
it
would
be rnore diflicult
to
solve. And you'll probably benefit frorn this book even if you can solve
this problern within an hour!