The Art and Craft of Problem Solving

58 CHAPTER 2 STRATEGIES FOR INVESTIGATING PROBLEMS

the planets are not touching each other.) It is easy to check that if n = 2, the total

private area is 4nR^2 , which is just the total area of one planet. What can you say about

n = 3? Other values of n?

Partial Solution: A bit of experimentation convinces us that if n = 3, the total

private area is also equal to the total area of one planet. Playing around with larger

n suggests the same result. We conjecture that the total private area is always equal

exactly to the area of one planet, no matter how the planets are situated. This appears

to be a nasty problem in solid geometry, but must it be? The notions of "private"

and "public" seem to be linked with a sort of duality; perhaps the problem is really

not geometric but logical. We need some "notation." Let us assume that there is a

universal coordinate system, such as longitude and latitude, so that we can refer to the

"same" location on any planet. For example, if the planets were little balls floating in

a room, the location "north pole" would mean the point on a planet that is closest to

the ceiling.

Given such a universal coordinate system, what can we say about a planet P that

has a private point at location x? Without loss of generality, let x be at the "north

pole." Clearly, the centers of all the other planets must lie on the south side of the

P's "equatorial" plane. But that renders the north poles of these planets public, for

their north poles are visible from a point in the southern hemisphere of P (or from the

southern hemisphere of any planet that lies between). In other words, we have shown

pretty easily that

If location x is private on one planet, it is public on all the other planets.

After this nice disc overy, the penultimate step is clear: to prove that

Given any location x, it must be private on some planet.^8

We leave this as an exercise (problem?) for the reader.

The above examples just scratched the surface of the vast body of crossover ideas.

While the concept of reformulating a problem is strategic, its implementation is tacti­

cal, frequently requiring specialized knowledge. We will discuss several other cross­

over ideas in detail in Chapter 4.

Change Your Point of View

Changing the point of view is just another manifestation of peripheral vision. Some­

times a problem is hard only because we choose the "wrong" point of view. Spending

a few minutes searching for the "natural" point of view can pay big dividends. Here is

a classic example.

Example 2.4.6 A person dives from a bridge into a river and swims upstream through

the water for 1 hour at constant speed. She then turns around and swims downstream

through the water at the same rate of speed. As the swimmer passes under the bridge, a

(^8) The sophisticated reader may observe that we are glossing over a technicality: there may be exceptional
points that do not obey this rule. However, these points form a set of measure zero, which will not affect the
result.