62 CHAPTER 3 TACTICS FOR SOLVING PROBLEMS
using) order can quickly simplify such problems. Consequently, we will begin by
studying problem-solving tactics that help us find or impose order where there seem
ingly is none. The most dramatic form of order is our first topic, symmetry.
Symmetry involves finding or imposing order in a concrete way, for example, by
reflection. Other tactics find or exploit order, but in more abstract, almost "metaphor
ical" ways. We shall discuss three such methods of "pseudosymmetry." All of them
rely on very simple observations that sometimes yield amazingly useful information.
The first tactic, the extreme principle, which we encountered with the Affirmative
Action problem (Example 2.1. 9 on page 21), works by focusing on the largest and
smallest entities within a problem. Next, the pigeonhole principle stems from the
nearly vacuous observation that if you have more guests than spare rooms, some of the
guests will have to share rooms. Our final tactic, the concept of invariants, shows how
much information comes your way when you restrict your attention to a narrow aspect
of your problem that does not change (such as parity). This is an extremely powerful
idea, fundamental to mathematics, that underlies many seemingly different tactics and
tools.3.1 Symmetry
We all have an intuitive idea of symmetry; for example, everyone understands that
circles are symmetrical. It is helpful, however, to define symmetry in a formal way, if
only because this will expand our notion of it. We call an object symmetric if there are
one or more non-trivial "actions" that leave the object unchanged. We call the actions
that do this the symmetries of the object. I
We promised something formal, but the above definition seems pretty vague. What
do we mean by an "action"? Almost anything! We are being deliberately vague,
because our aim is to let you see symmetry in as many situations as possible. Here are
a few examples.Example 3.1.1 A square is symmetric with respect to reflection about a diagonal. The
reflection is one of the several symmetries of the square. Other symmetries include
rotation clockwise by 90 degrees and reflection about a line joining the midpoints of
two opposite sides.Example 3.1. 2 A circle has infinitely many symmetries, for example, rotation clock
wise by a degrees for any a.Example 3.1. 3 The doubly infinite sequence... ,3, 1,4,3,1,4,3, 1,4, ...is symmetrical with respect to the action "shift everything three places to the right (or
left)."1 We deliberately avoid the language of transformations and automorphisms that would be demanded by a
mathematically precise definition.