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2 1 The How, When, and Why of Mathematics
Second. “Devising a plan.” How will you attack the problem? At this point, you
understand what must be done (because you have completed Step 1). Have you seen
something like it before? If you haven’t looked over class notes, haven’t read the
text, or haven’t done the previous homework assignments, the odds are slim that
you have seen anything that will be helpful. Do all that first. Look over the text with
the problem in mind, read over your notes with the problem in your head, look at
previous exercises and theorems that sound similar. Maybe you can use some of
the ideas in the proof of a theorem, or maybe you can use a previous homework
problem. Mathematics builds on itself and the problems in the text will also. If you
are truly stuck, try to answer a simpler, similar question. Once you decide on a
method of approach, try it out.
Third. “Carrying out the plan.” Solve the problem. Look at your solution. Is each
sentence true? Sometimes it is difficult to catch an error right after you have “found
a solution.” Put the problem down and come back to it a few hours later. Is each
sentence still true?
Fourth. “Looking back.” P
´
olya suggests checking the result and the argument, or
even looking for a different proof. If you are allowed (check with your teacher), one
really good way to check a proof is to give it to someone else. You can present it
to friends. Even if they don’t understand a word you are saying, sometimes saying
it out loud in a coherent manner will allow you to recognize an error you can’t spot
when you are reading. If you are permitted to work together, switch proofs and ask
your partner for criticism of your proof.
When you are convinced that your argument is correct, it is time to write up a
correct and neat solution to the problem.
Here is an example of the P
´
olya method at work in mathematics; we will decipher
a message. A cipher is a system that is used to hide the meaning of a message by
replacing the letters of the alphabet by other letters or symbols.
Exercise 1.1. The following message is encoded by a shift of the alphabet; that is,
every letter is replaced by another one that has been shifted n places further down
the alphabet. Once we reach the end of the alphabet, we start over. For instance, if n
were 7, we would make the replacements a → h, b → i, ... , s → z, t → a, .. .. Now
the exercise: What does the message below say?
PDEO AJYKZEJC WHCKNEPDI EO YWHHAZ W YWAOWN YELDAN.
EP EO RANU AWOU PK XNAWG, NECDP?
Let’s use the ideas from P
´
olya’s list to solve this. If you have solved problems
like this before, it might be a better exercise for you to try on your own to see how
this fits P
´
olya’s method before you read on.
1. “Understanding the problem.” Each sequence of letters with no blank space be-
tween the letters represents one word. Each letter is shifted by the same number
of places: namely, n. So n is the unknown in this problem and it is what we need
to find. Once we know the value of n, we can decipher the whole message. In
addition, once we know the meaning of one letter, we can find the value for n.