2.1 PSYCHOLOGICAL STRATEGIES 17There is a moral to the story, of course. Most people, when confronted with this
problem, immediately declare that it is impossible. Good problem solvers do not, how
ever. Remember, there is no time pressure. It might feel good to quickly "dispose" of a
problem, by either solving it or declaring it to be unsolvable, but it is far better to take
one's time to understand a problem. Avoid immediate declarations of impossibility;
they are dishonest.
We solved this problem by using two strategic principles. First, we used the psy
chological strategy of cultivating an open, optimistic attitude. Second, we employed
the enjoyable strategy of making the problem easier. We were lucky, for it turned out
that the original problem was almost immediately equivalent to the modified easier
version. That happened for a mathematical reason: the problem was a "topological"
one. This trick of mutating a diagram into a "topologically equivalent" one is well
worth remembering. It is not a strategy, but rather a tool, in our language.
Creativity
Most mathematicians are "Platonists," believing that the totality of their subject al
ready "exists" and it is the job of human investigators to "discover" it, rather than
create it. To the Platonist, problem solving is the art of seeing the solution that is al
ready there. The good problem solver, then, is highly open and receptive to ideas that
are floating around in plain view, yet invisible to most people.
This elusive receptiveness to new ideas is what we call creativity. Observing it
in action is like watching a magic show, where wonderful things happen in surprising,
hard-to-explain ways. Here is an example of a simple problem with a lovely, unex
pected solution, one that appeared earlier as Problem 1. 3. 1 on page 7. Please think
about the problem a bit before reading the solution!
Example 2.1.2 A monk climbs a mountain. He starts at 8AM and reaches the summit
at noon. He spends the night on the summit. The next morning, he leaves the summit at
8 AM and descends by the same route that he used the day before, reaching the bottom
at noon. Prove that there is a time between 8AM and noon at which the monk was at
exactly the same spot on the mountain on both days. (Notice that we do not specify
anything about the speed that the monk travels. For example, he could race at 1000
miles per hour for the first few minutes, then sit still for hours, then travel backward,
etc. Nor does the monk have to travel at the same speeds going up as going down.)
Solution: Let the monk climb up the mountain in whatever way he does it. At
the instant he begins his descent the next morning, have another monk start hiking up
from the bottom, traveling exactly as the first monk did the day before. At some point,
the two monks will meet on the trail. That is the time and place we want! _
The extraordinary thing about this solution is the unexpected, clever insight of
inventing a second monk. The idea seems to come from nowhere, yet it instantly
resolves the problem, in a very pleasing way. (See page 53 for a more "conventional"
solution to this problem.)
That's creativity in action. The natural reaction to seeing such a brilliant, imagi
native solution is to say, "Wow! How did she think of that? I could never have done