8 I The How,When,and Why of Mathematics Spotlight:George Polya Gyorgy Polya(1887-1985),referred to as George Polya in his later years,was born and raised in Hungary.He studied in Vienna and in Budapest,where he received his doctorate in 1912.One of his influential teachers was Leopold Fejer.In his book [85, p.39].Polya refers to Fejer as"an inspiring teacher who had a great deal of influence on Hungarian mathematicians of the time."The two primary places where Polya taught were the Eidgenossische Technische Hochschule(ETH)in Zurich,Switzer- land,and Stanford University in Palo Alto,California. Though Polya's mother tongue was Hungarian,he worked in the Swiss-German- speaking part of Switzerland and he spoke French with his wife from Neuchatel,a city in the French-speaking part of Switzerland.In school he also learned Latin and Greek.(See [85,p.11].)Polya later emigrated to the United States where he taught and lectured in English.He published mathematical papers in Hungarian,German, French,English,Italian,and Danish. Polya contributed to original research in probability,geometry,number theory, real and complex analysis,graph theory,combinatorics,and mathematical physics. His name is connected to many mathematical ideas and constructions.To name just a few of his achievements,we mention that in probability there is a Polya distribution and he is credited with introducing the idea and the term of"random walk."But Polya was not only recognized as an excellent scholar of mathematics,he was also an excellent teacher of mathematics.His heuristic approach to problem solving is outlined in How to Solve It.This book had a profound influence on the teaching of mathematics.It has sold over one million copies and is translated into over 20 languages.Records kept by the ETH in Zuirich show that Polya was the advisor of 14 thesis students there and,according to [78],he was the advisor of 9 more students at Stanford. The Mathematical Association of America(MAA)gives an annual Polya award. According to the MAA website,"This award,established in 1976,is named after the renowned teacher and writer,and is given for articles of expository excellence published in the College Mathematics Journal." To learn more about George Polya and his approach to problem solving,we rec- ommend reading his book How to Solve It,[84],the picture book [85](which con- tains a short biography),or consulting the more in-depth account of Polya's life [4], written by his former student at Stanford.The article [110]is based on interviews with Polya and appeared in an issue of Mathematics Magazine entirely devoted to Polya and his work. Problems Problem 1.1.Here is a problem intended to help you work through"the list."After this,you are on your own
8 1 The How, When, and Why of Mathematics Spotlight: George Polya ´ Gyorgy P ¨ olya (1887–1985), referred to as George P ´ olya in his later years, was born ´ and raised in Hungary. He studied in Vienna and in Budapest, where he received his doctorate in 1912. One of his influential teachers was Leopold Fejer. In his book [85, ´ p. 39], Polya refers to Fej ´ er as “an inspiring teacher who had a great deal of influence ´ on Hungarian mathematicians of the time.” The two primary places where Polya ´ taught were the Eidgenossische Technische Hochschule (ETH) in Z ¨ urich, Switzer- ¨ land, and Stanford University in Palo Alto, California. Though Polya’s mother tongue was Hungarian, he worked in the Swiss-German- ´ speaking part of Switzerland and he spoke French with his wife from Neuchatel, a ˆ city in the French-speaking part of Switzerland. In school he also learned Latin and Greek. (See [85, p. 11].) Polya later emigrated to the United States where he taught ´ and lectured in English. He published mathematical papers in Hungarian, German, French, English, Italian, and Danish. Polya contributed to original research in probability, geometry, number theory, ´ real and complex analysis, graph theory, combinatorics, and mathematical physics. His name is connected to many mathematical ideas and constructions. To name just a few of his achievements, we mention that in probability there is a Polya distribution ´ and he is credited with introducing the idea and the term of “random walk.” But Polya was not only recognized as an excellent scholar of mathematics, he was also ´ an excellent teacher of mathematics. His heuristic approach to problem solving is outlined in How to Solve It. This book had a profound influence on the teaching of mathematics. It has sold over one million copies and is translated into over 20 languages. Records kept by the ETH in Zurich show that P ¨ olya was the advisor of ´ 14 thesis students there and, according to [78], he was the advisor of 9 more students at Stanford. The Mathematical Association of America (MAA) gives an annual Polya award. ´ According to the MAA website, “This award, established in 1976, is named after the renowned teacher and writer, and is given for articles of expository excellence published in the College Mathematics Journal.” To learn more about George Polya and his approach to problem solving, we rec- ´ ommend reading his book How to Solve It, [84], the picture book [85] (which contains a short biography), or consulting the more in-depth account of Polya’s life [4], ´ written by his former student at Stanford. The article [110] is based on interviews with Polya and appeared in an issue of ´ Mathematics Magazine entirely devoted to Polya and his work. ´ Problems Problem 1.1. Here is a problem intended to help you work through “the list.” After this, you are on your own
I The How,When,and Why of Mathematics 9 Find a word(written in standard capital letters)that is unchanged when reflected in a horizontal line and in a vertical line.The word must appear in a dictionary(in a language of your choice)in order to be a valid solution. 1."Understanding the problem."We need to find a word.We are given information about the letters that make up this word.There are two conditions:Two different reflections should not alter the word. Try these two reflections on a word,say on SOLUTION,to make sure you un- derstand the problem. 2."Devising a plan."We have to find the connection between what we are given and what we have to find. Which letters of the alphabet satisfy each of the two conditions? Both conditions? Find a word that is not changed if it is reflected in a horizontal line. Find a word that is not changed if it is reflected in a vertical line. Formulate the exact conditions for this exercise;that is,state the letters that can be used and how they must be arranged. 3."Carrying out the plan."Find a word that satisfies the conditions given above. 4.“Looking back.”Are there other solutions? Problem 1.2.Find a word (written in standard capital letters)that reads the same forward and backward and is still the same forward and backward when rotated around its center 180.Your solution needs to appear in a standard dictionary of some language. Problem 1.3.Solve the following anagrams.The first three are places (in the ge- ographical sense),and the fourth is a place in which you might live.All can be rearranged to form a single word. (a)NOVA CURVE: (b)NINE SLAP NAVY: (c)I HELD A HIP PAL: (d)DIRTY ROOM. Note:You may have to find out exactly what an anagram is.This is part of Polya's first point on the list. Problem 1.4.Suppose n teams play in a single game elimination tournament.How many games are played? An example of such tournaments are the various categories of the U.S.Open tennis tournament;for example,women's singles. Note:Pay special attention to the first entry of Polya's list:"Is it possible to satisfy the condition?" Problem 1.5.Suppose you are all alone in a strange house.There are seven identical closed doors.The bathroom is behind exactly one of them.Is it more likely,less likely,or equally likely that you find the bathroom on the first try than on the third try?Why?
1 The How, When, and Why of Mathematics 9 Find a word (written in standard capital letters) that is unchanged when reflected in a horizontal line and in a vertical line. The word must appear in a dictionary (in a language of your choice) in order to be a valid solution. 1. “Understanding the problem.” We need to find a word. We are given information about the letters that make up this word. There are two conditions: Two different reflections should not alter the word. Try these two reflections on a word, say on SOLUTION, to make sure you understand the problem. 2. “Devising a plan.” We have to find the connection between what we are given and what we have to find. Which letters of the alphabet satisfy each of the two conditions? Both conditions? Find a word that is not changed if it is reflected in a horizontal line. Find a word that is not changed if it is reflected in a vertical line. Formulate the exact conditions for this exercise; that is, state the letters that can be used and how they must be arranged. 3. “Carrying out the plan.” Find a word that satisfies the conditions given above. 4. “Looking back.” Are there other solutions? Problem 1.2. Find a word (written in standard capital letters) that reads the same forward and backward and is still the same forward and backward when rotated around its center 180◦. Your solution needs to appear in a standard dictionary of some language. Problem 1.3. Solve the following anagrams. The first three are places (in the geographical sense), and the fourth is a place in which you might live. All can be rearranged to form a single word. (a) NOVA CURVE; (b) NINE SLAP NAVY; (c) I HELD A HIP PAL; (d) DIRTY ROOM. Note: You may have to find out exactly what an anagram is. This is part of Polya’s ´ first point on the list. Problem 1.4. Suppose n teams play in a single game elimination tournament. How many games are played? An example of such tournaments are the various categories of the U. S. Open tennis tournament; for example, women’s singles. Note: Pay special attention to the first entry of Polya’s list: “Is it possible to ´ satisfy the condition?” Problem 1.5. Suppose you are all alone in a strange house. There are seven identical closed doors. The bathroom is behind exactly one of them. Is it more likely, less likely, or equally likely that you find the bathroom on the first try than on the third try? Why?
10 I The How,When,and Why of Mathematics Problem 1.6.The following message is encoded using a shifted alphabet just as in Exercise 1.1.(Of course,the shift number n is not the same as in the exercise!)What does the message say? RDSXCVIWTDGNXHUJCLTLXAAATPGCBDGTPQDJIXIAPITG Problem 1.7.Give a detailed description of all points in three-space that are equidis- tant from the x-axis and the yz-plane.Once you decide on the answer,write the solution up carefully.Pay particular attention to your notation. Problem 1.8.The following is a classic problem in mathematics.Though there are many variations of this problem,the standard one is the following. You are given 12 coins that appear to be identical.However,one of the coins is counterfeit,and the weight of this coin is slightly different than that of the other 11. Using only a two-pan balance,what is the smallest number of weighings you would need to find the counterfeit coin?(Think about a simpler,similar problem.) (See I.Peterson's website [82]for a discussion of this problem.) Problem 1.9.Let n be an odd integer.Prove that n3-n is divisible by 24. The following two problems are only appropriate if you took at least two semesters of calculus.Though you may have worked these before,the idea is to work them again paying close attention to the final presentation.Make sure you define all vari- ables.Use complete sentences,with proper punctuation. Problem 1.10.Find the volume of a spherical cap if the height is 2 m and the radius of the rim of the cap is 5 m. Problem 1.11.We have two circular right cylinders of radius 1 each.The axes of the two cylinders intersect at a right angle.Find the volume of the solid that both cylinders have in common. Problem 1.12.Shlomo Sureshot started the basketball season with a free throw shooting percentage of below 75%.By the end of the season he brought it up to above 75%.Must there have been a time in the season (after a free throw attempt) when his free throw percentage was exactly 75%? After having solved this problem and looking back at your solution,are there questions that you would like to answer?Can you answer them? Problem 1.13.An old Spanish treasure is hidden away in a magical box with integer dimensions;that is,height,width,and length are all of integer values if measured in "braza,"a unit of length.The volume of the box is 40 braza'.Though knowing the sum of the dimensions will not completely determine them,knowing(in addition) that the square front face is painted red will.What are the dimensions of the box?
10 1 The How, When, and Why of Mathematics Problem 1.6. The following message is encoded using a shifted alphabet just as in Exercise 1.1. (Of course, the shift number n is not the same as in the exercise!) What does the message say? RDSXCVIWTDGNXHUJCLTLXAAATPGCBDGTPQDJIXIAPITG Problem 1.7. Give a detailed description of all points in three-space that are equidistant from the x-axis and the yz-plane. Once you decide on the answer, write the solution up carefully. Pay particular attention to your notation. Problem 1.8. The following is a classic problem in mathematics. Though there are many variations of this problem, the standard one is the following. You are given 12 coins that appear to be identical. However, one of the coins is counterfeit, and the weight of this coin is slightly different than that of the other 11. Using only a two-pan balance, what is the smallest number of weighings you would need to find the counterfeit coin? (Think about a simpler, similar problem.) (See I. Peterson’s website [82] for a discussion of this problem.) Problem 1.9. Let n be an odd integer. Prove that n3 −n is divisible by 24. The following two problems are only appropriate if you took at least two semesters of calculus. Though you may have worked these before, the idea is to work them again paying close attention to the final presentation. Make sure you define all variables. Use complete sentences, with proper punctuation. Problem 1.10. Find the volume of a spherical cap if the height is 2 m and the radius of the rim of the cap is 5 m. Problem 1.11. We have two circular right cylinders of radius 1 each. The axes of the two cylinders intersect at a right angle. Find the volume of the solid that both cylinders have in common. Problem 1.12. Shlomo Sureshot started the basketball season with a free throw shooting percentage of below 75%. By the end of the season he brought it up to above 75%. Must there have been a time in the season (after a free throw attempt) when his free throw percentage was exactly 75%? After having solved this problem and looking back at your solution, are there questions that you would like to answer? Can you answer them? Problem 1.13. An old Spanish treasure is hidden away in a magical box with integer dimensions; that is, height, width, and length are all of integer values if measured in “braza,” a unit of length. The volume of the box is 40 braza3. Though knowing the sum of the dimensions will not completely determine them, knowing (in addition) that the square front face is painted red will. What are the dimensions of the box?
I The How,When,and Why of Mathematics 11 Tips on Doing Homework Your instructor will probably ask you to work many of the exercises and problems in this text.If there is one thing mathematicians agree on,it is that you learn math- ematics by doing it.Here are some tips on how to get started. Make sure you know what the rules are.Some instructors do not want you to get help from someone else.Other instructors encourage working together in groups. Ask,if you are not clear about the policy. If you are permitted to work together,form a study group.A small group of two to four people usually works best.Get together on a regular basis and discuss the assigned problems. Read the questions carefully.If there is a term that you do not know,look it up. Before you get started,read over the text and the notes from class,paying partic- ular attention to definitions,theorems,and previous exercises.It isn't unusual to spend several hours on a single problem at this point.Doing mathematics means pondering a problem for hours,days,weeks,even years(though we have tried not to pose problems that will take you years to solve).Working two hours on one problem,thinking about it as you go through your day and then spending another two hours on it the next day is fairly common practice for students at this level. Once you have read over the text,looked over the relevant definitions,worked through the examples,and tried to solve the problem,you will be well on your way toward understanding the problem.If you can't get started,at least you will know which questions to ask.Seek help from your instructor or other students (if your instructor allows this). Once you have a solution to a problem,look at it critically.Check that it is correct. Put it down.Come back to it later.Do you still understand everything?Is it still correct?(As you can imagine,this is very important.)Can you simplify it?If you work with someone else,have them read it over.Never hand in your first draft of a solution to a problem. Writing a solution means convincing a reader that the result is correct.There can be no gaps or errors.Explain each step-don't assume that the reader knows what you are thinking.Keep a reader in mind as you write,and remember that the instructor or anyone else who already knows the solution is not really your target audience.Though that may be the person for whom the solution is intended,it is your job to convince the reader that each step in your solution is correct.Perhaps a better audience to keep in mind is someone who knows the material from the class,but not the solution to the problem. Write up your final solution very carefully and neatly.The reader shouldn't find him-or herself proving things for you-you should do that for him or her.Staple pages together so that the reader may have the pleasure of reading your proof in the correct order and its entirety
1 The How, When, and Why of Mathematics 11 Tips on Doing Homework Your instructor will probably ask you to work many of the exercises and problems in this text. If there is one thing mathematicians agree on, it is that you learn mathematics by doing it. Here are some tips on how to get started. • Make sure you know what the rules are. Some instructors do not want you to get help from someone else. Other instructors encourage working together in groups. Ask, if you are not clear about the policy. • If you are permitted to work together, form a study group. A small group of two to four people usually works best. Get together on a regular basis and discuss the assigned problems. • Read the questions carefully. If there is a term that you do not know, look it up. • Before you get started, read over the text and the notes from class, paying particular attention to definitions, theorems, and previous exercises. It isn’t unusual to spend several hours on a single problem at this point. Doing mathematics means pondering a problem for hours, days, weeks, even years (though we have tried not to pose problems that will take you years to solve). Working two hours on one problem, thinking about it as you go through your day and then spending another two hours on it the next day is fairly common practice for students at this level. • Once you have read over the text, looked over the relevant definitions, worked through the examples, and tried to solve the problem, you will be well on your way toward understanding the problem. If you can’t get started, at least you will know which questions to ask. Seek help from your instructor or other students (if your instructor allows this). • Once you have a solution to a problem, look at it critically. Check that it is correct. Put it down. Come back to it later. Do you still understand everything? Is it still correct? (As you can imagine, this is very important.) Can you simplify it? If you work with someone else, have them read it over. Never hand in your first draft of a solution to a problem. • Writing a solution means convincing a reader that the result is correct. There can be no gaps or errors. Explain each step—don’t assume that the reader knows what you are thinking. Keep a reader in mind as you write, and remember that the instructor or anyone else who already knows the solution is not really your target audience. Though that may be the person for whom the solution is intended, it is your job to convince the reader that each step in your solution is correct. Perhaps a better audience to keep in mind is someone who knows the material from the class, but not the solution to the problem. • Write up your final solution very carefully and neatly. The reader shouldn’t find him- or herself proving things for you—you should do that for him or her. Staple pages together so that the reader may have the pleasure of reading your proof in the correct order and its entirety
Chapter 2 Logically Speaking "I know what you're thinking about,"said Tweedledum;"but it isn't so,nohow.""Contrari- wise."continued Tweedledee,"if it was so,it might be:and if it were so.it would be;but as it isn't,it ain't.That's logic."-Lewis Carroll,[17,p.139] Suppose your friend tells you that Mr.Hamburger is German or Swiss.You hap- pen to know that Mr.Hamburger is not Swiss.Using your powers of reasoning,you decide that Mr.Hamburger is German.Note that this argument can be generalized, because it doesn't really depend on Mr.Hamburger being Swiss or German.If your friend said that"A or B is true"and you happened to know that"B is not true," you would conclude that"A is true."This is an example of a valid argument.Now suppose your friend tells you that Mr.French eats only pickles on Wednesday,and only chocolate on Monday.You know that Mr.French is eating chocolate that day. Now what can you say?While you may conclude that Mr.French has odd eating habits,you would not have used a logically valid argument to do so.In this example, there is really only one thing you can conclude.We'll return to this at the end of this chapter. In order to understand an argument,we must be able to read and comprehend the sentences that compose it.We need to be able to tell whether the sentences in our argument are true or false,and whether they follow logically from the previous ones.So now for a definition.A statement is a sentence that is true or false,but not both."Two is not a prime number"is an example of a (false)statement."Do you love me?"is not a statement.Below are some examples and some nonexamples of statements.These will be your first examples of nonexamples. Exercise 2.1.Which of the sentences below are statements and which are not? (a)It is raining outside (b)The professor of this class is a woman. (c)Two plus two is five. (dX+6=0. (e)Seven is a prime number. (f)All odd numbers are prime. U.Daepp and P.Gorkin,Reading.Writing,and Proving:A Closer Look at Mathematics, 13 Undergraduate Texts in Mathematics,DOI 10.1007/978-1-4419-9479-0_2, Springer Science+Business Media,LLC 2011
Chapter 2 Logically Speaking “I know what you’re thinking about,” said Tweedledum; “but it isn’t so, nohow.” “Contrariwise,” continued Tweedledee, “if it was so, it might be; and if it were so, it would be; but as it isn’t, it ain’t. That’s logic.”—Lewis Carroll, [17, p. 139] Suppose your friend tells you that Mr. Hamburger is German or Swiss. You happen to know that Mr. Hamburger is not Swiss. Using your powers of reasoning, you decide that Mr. Hamburger is German. Note that this argument can be generalized, because it doesn’t really depend on Mr. Hamburger being Swiss or German. If your friend said that “A or B is true” and you happened to know that “B is not true,” you would conclude that “A is true.” This is an example of a valid argument. Now suppose your friend tells you that Mr. French eats only pickles on Wednesday, and only chocolate on Monday. You know that Mr. French is eating chocolate that day. Now what can you say? While you may conclude that Mr. French has odd eating there is really only one thing you can conclude. We’ll return to this at the end of this chapter. In order to understand an argument, we must be able to read and comprehend the sentences that compose it. We need to be able to tell whether the sentences in our argument are true or false, and whether they follow logically from the previous ones. So now for a definition. A statement is a sentence that is true or false, but not both. “Two is not a prime number” is an example of a (false) statement. “Do you love me?” is not a statement. Below are some examples and some nonexamples of statements. These will be your first examples of nonexamples. Exercise 2.1. Which of the sentences below are statements and which are not? (a) It is raining outside. (b) The professor of this class is a woman. (c) Two plus two is five. (d) X +6 = 0. (e) Seven is a prime number. (f) All odd numbers are prime. © Springer Science+Business Media, LLC 2011 U. Daepp and P. Gorkin, Reading, Writing, and Proving: A Closer Look at Mathematics, 13 habits, you would not have used a logically valid argument to do so. In this example, Undergraduate Texts in Mathematics, DOI 10.1007/978-1-4419-9479-0_2