The Art and Craft of Problem Solving

64 CHAPTER 3 TACTICS FOR SOLVING PROBLEMS

The small square obviously has half the area of the larger! -

The simplest geometric symmetries are rotational and reflectional. Always check

to see if rotations or reflections will impose order on your problem. The next example

shows the power of imposing reflectional symmetry.

Example 3.1. 5 Your cabin is two miles due north of a stream that runs east-west. Your

grandmother's cabin is located 12 miles west and one mile north of your cabin. Every

day, you go from your cabin to Grandma's, but first visit the stream (to get fresh water

for Grandma). What is the length of the route with minimum distance?

Solution: First, draw a picture! Label your location by Y and Grandma's by

G. Certainly, this problem can be done with calculus, but it is very ugly (you need

to differentiate the sum of two radicals, for starters). The problem appears to have no

symmetry in it, but the stream is practically begging you to reflect across it! Draw a

sample path (shown below as Y A followed by AG) and look at its reflection. We call

the reflections of your house and Grandma's house y' and G', respectively.

G

--------�-=�------��--------------Stream

G'

While you are carrying water to Grandma, your duplicate in an alternate universe

is doing the same, only south of the stream. Notice that AG = AG' , so the length of

your path would be unchanged if you visited the reflected Grandma instead of the real

one. Since the shortest distance between two points is the straight line Y BG', the op­

timal path will be Y B followed by BG. Its length is the same as the length of Y BG',

which is just the hypotenuse of a right triangle with legs 5 and 12 miles. Hence our

answer is 13 miles. _

When pondering a symmetrical situation, you should always focus briefly on the

"fixed" objects that are unchanged by the symmetries. For example, if something is

symmetric with respect to reflection about an axis, that axis is fixed and worthy of

study (the stream in the previous problem played that role). Here is another example,

a classic problem that exploits rotational symmetry along with a crucial fixed point.