2 I 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."Polya 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 Polya 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 were7,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 Polya'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 Polya'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.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.” Polya 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 Polya 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 Polya’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 Polya’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