"Mr. Northrop writes well and simply. Every so often he will illuminate his discussion with an amusing example. While reading a discussion of topology, the reviewer learned how to remove his vest from beneath his jacket. It works every time." — The New York Times
"Mr. Northrop writes well and simply. Every so often he will illuminate his discussion with an amusing example. While reading a discussion of topology, the reviewer learned how to remove his vest from beneath his jacket. It works every time." — The New York Times
Riddles in Mathematics: A Book of Paradoxes
288
Riddles in Mathematics: A Book of Paradoxes
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"Mr. Northrop writes well and simply. Every so often he will illuminate his discussion with an amusing example. While reading a discussion of topology, the reviewer learned how to remove his vest from beneath his jacket. It works every time." — The New York Times
Product Details
| ISBN-13: | 9780486795003 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 05/15/2014 |
| Series: | Dover Recreational Math |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 288 |
| File size: | 15 MB |
| Note: | This product may take a few minutes to download. |
| Age Range: | 14 - 18 Years |
About the Author
Read an Excerpt
Riddles In Mathematics
A Book of Paradoxes
By Eugene P. Northrop
Dover Publications, Inc.
Copyright © 1972 Marion L. NorthropAll rights reserved.
ISBN: 978-0-486-79500-3
CHAPTER 1
What Is a Paradox?
Two fathers and two sons leave town. This reduces the population of the town by three. False? No, true—provided the trio consists of father, son, and grandson.
A bookworm starts at the outside of the front cover of volume I of a certain set of books and eats his way to the outside of the back cover of volume III. If each volume is one inch thick, he must travel three inches in all. True? No, false. A moment's study of the accompanying figure shows that he has only to make his way through volume II—a distance of one inch.
A man says, "I am lying." Is his statement true? If so, then he is lying, and his statement is false. Is his statement false? If so, then he is lying, and his statement is true.
The dictionaries define an island as "a body of land completely surrounded by water" and a lake as "a body of water completely surrounded by land." But suppose the northern hemisphere were all land, and the southern hemisphere all water. Would you call the northern hemisphere an island, or would you call the southern hemisphere a lake?
It is of such brain-twisters as these that this book is composed. There are paradoxes for everyone—from the person who left mathematics behind in school (or who was left behind in school by mathematics) to the professional mathematician, who is still bothered by such a problem as that of the liar.
We shall use the word "paradox," by the way, in the sense in which it is used in these examples. That is to say, a paradox is anything which offhand appears to be false, but is actually true; or which appears to be true, but is actually false; or which is simply self-contradictory. From time to time it may appear that we are straying from this meaning. But be patient—what seems crystal- clear to you may leave the next person completely confused.
* * *
If you are among those who at this point are saying, "But we thought this book had to do with mathematical paradoxes—how about it?" then stay with us for a moment. If you are not interested in the answer to this question, you may as well skip to the next chapter.
A closer look at the difficulties encountered in our first examples will show that they are simple cases of very real difficulties encountered not only by the student of mathematics, but by the mature mathematician as well.
In the problem concerning fathers and sons, we find ourselves searching here and there for some instance in which the conditions of the problem will be fulfilled. It seems at first as though such an instance cannot possibly exist—common sense and intuition are all against it. But suddenly, there it is—as simple a solution as can be. This sort of thing happens time and again in mathematical research. The mathematician, working on the development of some theory or other, is suddenly confronted with a set of conditions which appear to be highly improbable. He begins looking for an example to fit the conditions, and it may be days, or weeks, or even longer, before he finds one. Frequently the solution of his difficulty is as simple as was ours—the kind of thing that makes him wonder why he hadn't thought of it before.
The problem of the bookworm's journey is a nice example of the way in which reason can be led astray by hasty judgment. The false conclusion is reached through failure to investigate carefully all aspects of the problem. There are many specimens of this sort—much more subtle ones, to be sure—in the literature of mathematics. A number of them enjoyed careers lasting many years before some doubting mathematician finally succeeded in discovering the trouble.
The case of the self-contradicting liar is but one of a whole string of logical paradoxes of considerable importance. Invented by the early Greek philosophers, who used them chiefly to confuse their opponents in debate, they have in more recent times served to bring about revolutionary changes in ideas concerning the nature and foundations of mathematics. In a later chapter we shall have more to say about problems of this kind.
The island-and-lake problem, which had to do with definitions and reasoning from definitions, is really typical of the development of any mathematical theory. The mathematician first defines the objects with which he is going to work—numbers, or points, or lines, or even just "elements" of an unspecified nature. He then lays down certain laws—"axioms," he calls them, or "postulates"—which are to govern the behavior of the objects he has defined. On this foundation he builds, through a series of logical arguments, a whole structure of mathematical propositions, each one resting on the conclusions established before it. He is not interested, by the way, in the truth of his definitions or axioms, but asks only that they be consistent, that is, that they lead to no real contradiction in the propositions (such, for example, as the contradiction in the problem of the liar). Bertrand Russell, in his Mysticism and Logic, has put what we are trying to say in the following words: "Pure mathematics consists entirely of assertions to the effect that if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is of which it is supposed to be true.... Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true." How is that, by the way, for a paradox?
CHAPTER 2A Few Simple Brain-Teasers
Many of the anecdotes and problems of this chapter are fairly well known. All of them have probably appeared in print in some form or other and at some time or other, and a few are so common that they can be found in almost any book on mathematical puzzles and games. It is next to useless to try to trace them to their original sources—most of them, like Topsy, "just growed."
* * *
We shall begin with a couple of lessons in geography. The first concerns a man who, you will say, must have been a crank. He designed a square house with windows on all four sides, each window having a view to the south. No bay windows (which would take care of three sides) or anything of that sort. Now how on earth can this be done? Where on earth would be more to the point, for there is indeed only one place where such a house can be built. Does that give it away? You've got it—it's the North Pole, of course, from which any direction is south.
Without the foregoing discussion, the following problem strikes most people as quite paradoxical. A certain sportsman, experienced in shooting small game, was out on his first bear hunt. Suddenly he spotted a huge bear about a hundred yards due east of him. Seized with panic, the hunter ran—not directly away from the bear, but, in his confusion, due north. Having covered about a hundred yards, he regained his presence of mind, stopped, turned, and killed the bear—who had not moved from his original position—by shooting due south. Have you all the data clearly in mind? Very well, then; what color was the bear?
The same problem can be put in another, although perhaps less startling, form. Where can a man set out from his house, walk five miles due south, five miles due west, and five miles due north—and find himself back home?
* * *
Charles L. Dodgson, better known to the general public as Lewis Carroll, the author of Alice in Wonderland, is recognized by mathematicians and logicians as one of their own number. We are indebted to him for the following paradox, as well as for several others which appear in later parts of the book.
We can agree, can we not, that the better of two clocks is the one that more often shows the correct time? Now suppose we are offered our choice of two clocks, one of which loses a minute a day, while the other does not run at all. Which one shall we accept? Common sense tells us to take the one that loses a minute a day, but if we are to stick to our agreement, we shall have to take the one that doesn't run at all. Why? Well, the clock that loses a minute a day, once properly set, will have to lose 12 hours, or 720 minutes, before it is right again. And if it loses only a minute a day, it will take 720 days to lose 720 minutes. In other words, it is correct only once about every two years. But the clock that doesn't run at all is correct twice a day!
* * *
Apparently impossible results are frequently obtained through either too little attention to relevant details or too much attention to irrelevant ones. Let's look at a few problems of this kind. We shan't bother, by the way, to discuss their solutions here.
A scatter-brained young lady once went into a jewelry store, picked out a ring worth $5, paid for it, and left. She appeared at the store the next day and asked if she might exchange it for another. This time she picked out one worth $10, thanked the jeweler sweetly, and started to leave. He naturally demanded an additional $5. The young lady indignantly pointed out that she had paid him $5 the day before, that she had just returned to him a $5 ring, and that she therefore owed him nothing. Whereupon she stalked out of the store and left the jeweler wildly counting on his fingers.
Then there is the story of the young man who once found himself applying for a job. He told the manager that he thought he was worth $1500 a year to him. The manager apparently thought otherwise. "Look here," he said, "there are 365 days in the year. You sleep 8 hours a day, or a total of 122 days. That leaves 243. You rest 8 hours a day, or a total of 122 days. That leaves 121. You do no work for 52 Sundays. That leaves 69 days. You have half a day off on 52 Saturdays—a total of 26 days. That leaves 43. You have an hour off for lunch each day—a total of 15 days. That leaves 28. You have 2 weeks of vacation. That leaves 14 days. And then come Fourth of July, Labor Day, Thanksgiving, and Christmas. Do you think you're worth $1500 to me for 10 working days?"
A group of seven weary men once arrived at a small hotel and asked for accommodations for the night, specifying that they wanted separate rooms. The manager admitted that he had only six rooms left, but thought he might be able to put up his guests as they desired. He took the first man to the first room and asked one of the other men to stay there for a few minutes. He then took the third man to the second room, the fourth man to the third room, the fifth man to the fourth room, and the sixth man to the fifth room. Then he returned to the first room, got the seventh man, and showed him to the sixth room. Everyone was thus nicely taken care of. Or was he?
If that last problem is too simple, as indeed it is, here is another about travelers and lodgings. Three men registered at a hotel and asked for connecting rooms. They were told of an available suite for which the charge was $30, and went up to look it over. Finding it satisfactory, they agreed to take it, and each man gave a $10 bill to the bellhop who accompanied them. He went down to the office to turn the money over to the cashier, and was met by the manager with the news that there had been a mistake—the charge for the suite was $25 not $30. Consequently the bellhop was sent back with five $1 bills. On the way it occurred to him that $5 was going to be difficult to divide among three men, that the men did not know the actual cost of the rooms anyway, and that they would be glad of any return on their money. So he pocketed two of the $1 bills and returned one to each of the three men. Now each of the men paid $9. Three times $9 is $27. The bellhop had $2 in his pocket. $27 plus $2 is $29, and the men originally handed over $30. Where is that other dollar?
While we are on the subject of dollars, there is that very puzzling story having to do with foreign exchange. The governments of two neighboring countries—let's call them Northia and Southia—had an agreement whereby a Northian dollar was worth a dollar in Southia, and vice versa. But one day the government of Northia decreed that thereafter a Southian dollar was to be worth but ninety cents in Northia. The next day the Southian government, not to be outdone, decreed that thereafter a Northian dollar was to be worth but ninety cents in Southia. Now a bright young man lived in a town which straddled the border between the two countries. He went into a store on the Northian side, bought a ten-cent razor, and paid for it with a Northian dollar. He was given a Southian dollar, worth ninety cents there, in change. He then crossed the street, went into a Southian store, bought a ten-cent package of blades, and paid for it with the Southian dollar. There he was given a Northian dollar in change. When the young man returned home, he had his original dollar and his purchases. And each of the tradesmen had ten cents in his cash-drawer. Who, then, paid for the razor and blades?
* * *
One of the oldest paradoxes is that of the wealthy Arab who at death left his stable of seventeen beautiful horses to his three sons. He specified that the eldest was to have one half the horses, the next one third, and the youngest one ninth. The three young heirs were in despair, for they obviously could not divide seventeen horses this way without calling in the butcher. They finally sought the advice of an old and wise friend, who promised to help them. He arrived at the stable the next day, leading one of his own horses. This he added to the seventeen and directed the brothers to make their choices. The eldest took one half of the eighteen, or nine; the next, one third of the eighteen, or six; and the youngest, one ninth of the eighteen, or two. When all seventeen of the original horses had been chosen, the old man took his own horse and departed. The catch? It's in the father's stipulations. Either he was a poor arithmetician or he wanted to give his sons something to thank about. Any rate that fractions one half, one third, and one ninth do not add up to unity—as they should if nothing is to be left over—but to seventeen eighteenths.
* * *
A large business firm was once planning to open a new branch in a certain city, and advertised positions for three clerks. Out of a number of applicants the personnel manager selected three promising young men and addressed them in the following way: "Your salaries are to begin at the rate of $1000 per year, to be paid every half-year. If your work is satisfactory, and we keep you, your salaries will be raised. Which would you prefer, a raise of $150 per year or a raise of $50 every half-year?" The first two of the three applicants eagerly accepted the first alternative, but the third young man, after a moment's reflection, took the second. He was promptly put in charge of the other two. Why? Was it because the personnel manager liked his modesty and apparent willingness to save the company money? Not at all. As befitting his position, he actually received more salary than his companions. They had jumped to the conclusion that a raise of $50 every half-year was equivalent to a raise of $100 per year, but he had taken all of the conditions of the problem into consideration. He had lined up the two possibilities and had looked at the yearly salaries in this way:
150 raise yearly $50 raise half-year
1st year: $500 + $500 = $1000 $500 + $550 = $1050
2nd year: 575 + 575 = 1150 600 + 650 = 1250
3rd year: 650 + 650 = 1300 700 + 750 = 1450
4th year: 725 + 725 = 1450 800 + 850 = 1650
It was then immediately apparent to him that his salary in succeeding years would exceed theirs by $50, 100, 150, 200, ..., his raise each year exceeding theirs by $50. It was his alertness of mind, and not his modesty, that impressed his new employer.
* * *
Most people are easily confused by problems involving average rates of speed. Try this one on your friends.
A man drove his car 1 mile to the top of a mountain at the rate of 15 miles per hour. How fast must he drive 1 mile down the other side in order to average 30 miles per hour for the whole trip of 2 miles?
First let us look at it in this way: he would average 30 miles per hour for the whole trip if he drove the second mile at the rate of 45 miles per hour, for the average of 15 and 45 is (15+45)/2, or 30.
But now suppose we look at it in another way. Using our old friend, the relation "distance = rate X time," we note that the time required to drive 2 miles at the average rate of 30 miles per hour is 2/30 of an hour, or 4 minutes. Furthermore, the time required to drive 1 mile at the rate of 15 miles per hour is 1/15 of an hour, or again 4 minutes. In other words, our traveler must cover that second mile in 0 seconds flat!
Which of these results are we to accept? The second is the correct one, and shows that considerable care must be used in averaging rates. The average rate for any trip is always found by dividing the total distance by the total time. In our first analysis, if the man drives one mile at 15 miles per hour and a second mile at 45 miles per hour, the times for those two miles are 1/15 and 1/45 of an hour respectively, or 4/45 of an hour in all. His average rate is thus 2/ (4/45) or 22.5 miles per hour. This discussion should furnish a practical tip to those drivers who allow just so much time to get somewhere. They cannot average 50 miles per hour, for example, by going a certain number of miles at 40 miles per hour and the same number of miles at 60 miles per hour. On the other hand, they can average 50 by going 40 and 60 for the same number of hours. For if they maintain these respective rates for one hour each, they will have gone 100 miles in 2 hours.
With the help of the above discussion you ought to be able to pick out the flaws in the following two arguments. If not, you will find their solutions in the Appendix toward the end of the book.
(Continues...)
Excerpted from Riddles In Mathematics by Eugene P. Northrop. Copyright © 1972 Marion L. Northrop. Excerpted by permission of Dover Publications, Inc..
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