The Art and Craft of Problem Solving

3.1 SYMMETRY 65

Example 3.1. 6 Four bugs are situated at each vertex of a unit square. Suddenly, each
bug begins to chase its counterclockwise neighbor. If the bugs travel at 1 unit per
minute, how long will it take for the four bugs to crash into one another?

Solution: The situation is rotationally symmetric in that there is no one "distin­
guished" bug. If their starting configuration is that of a square, then they will always
maintain that configuration. This is the key insight, believe it or not, and it is a very
profitable one!

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: bug's path I

As time progresses, the bugs form a shrinking square that rotates counterclock­
wise. The center of the square does not move. This center, then, is the only "distin­
guished" point, so we focus our analysis on it. Many otherwise intractable problems
become easy once we shift our focus to the natural frame of reference; in this case
we should consider a radial frame of reference, one that rotates with the square. For
example, pick one of the bugs (it doesn't matter which one!), and look at the line
segment from the center of the square to the bug. This segment will rotate counter­
clockwise, and (more importantly) shrink. When it has shrunk to zero, the bugs will
have crashed into one another. How fast is it shrinking? Forget about the fact that the
line is rotating. From the point of view of this radial line, the bug is always traveling
at a 45 ° angle. Since the bug travels at unit speed, its radial velocity component is just

(^1) · cos 45° = V2/2 units per minute, i.e., the radial line shrinks at this speed. Since the

original length of the radial line was V2/2, it will take just 1 minute for the bugs to

crash. _

Here is a simple calculus problem that can certainly be solved easily in a more
conventional way. However, our method below illustrates the power of the Draw a
Picture strategy coupled with symmetry, and can be applied in many harder situations.

In
Example 3.1.7 Compute 102 cos^2 xdx in your head.

Solution: Mentally draw the sine and cosine graphs from 0 to 11r, and you will

notice that they are symmetric with respect to reflection about the vertical line x = i 1r.