Each chapter begins with a question about strings consisting of nothing more than two letters, and every such question raises intriguing problems to be explored and solved. Author Sherman Stein proceeds at a measured pace that permits readers to move through the chapters in a leisurely fashion, omitting none of the steps. His approach makes complex subjects — from topology to set theory to probability — both accessible and exciting.
Each chapter begins with a question about strings consisting of nothing more than two letters, and every such question raises intriguing problems to be explored and solved. Author Sherman Stein proceeds at a measured pace that permits readers to move through the chapters in a leisurely fashion, omitting none of the steps. His approach makes complex subjects — from topology to set theory to probability — both accessible and exciting.
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Overview
Each chapter begins with a question about strings consisting of nothing more than two letters, and every such question raises intriguing problems to be explored and solved. Author Sherman Stein proceeds at a measured pace that permits readers to move through the chapters in a leisurely fashion, omitting none of the steps. His approach makes complex subjects — from topology to set theory to probability — both accessible and exciting.
Product Details
| ISBN-13: | 9780486814117 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 07/19/2016 |
| Series: | Dover Books on Mathematics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| File size: | 6 MB |
About the Author
Read an Excerpt
Adventures in Mathematical Reasoning
By Sherman Stein
Dover Publications, Inc.
Copyright © 2001 Sherman SteinAll rights reserved.
ISBN: 978-0-486-81411-7
CHAPTER 1
The Needle and the Noodle
Georges Buffon (1707–1788) made his reputation with the publication of his multivolume Natural History, General and Particular, which brought order to much of what was known about the animal and mineral worlds. In an appendix totally unrelated to natural history, he includes a mainly mathematical work, titled Essay on Moral Arithmetic. One of the problems he discusses there concerns a needle dropped at random on a floor furnished with regularly spaced parallel lines.
"I suppose that, in a room where the floor is divided by parallel cracks, one throws a stick into the air. One gambler wagers that the stick will not cross any cracks. The other, on the contrary, wagers that it will cross some of them. What are the odds of winning for each of the gamblers? (One could play this game with a needle or a pin without a head.)"
This is known as the Buffon Needle problem.
The Needle
For convenience we will assume that all the lines (cracks) are the same distance apart, namely, the width of the slats, and that the length of the needle is the same as the distance between the cracks. We also assume that these lengths are 1 inch.
Buffon and the two gamblers want to know the likelihood that the needle will miss all the lines and also the likelihood that it will cross a line. Because the needle is not longer than the distance between the lines, it cannot cross two lines. (The case in which the needle lands perpendicular to the lines with its ends just touching the lines occurs so rarely that it will not affect our reasoning.) The typical possibilities are shown here.
We assume that the room is of infinite size. That is, the lines are infinitely long, and there is an infinite number of them. That way, our thinking will not be complicated by a border.
It may seem that the answer to Buffon's question will require a good deal of geometry. However, our reasoning will require none at all, in keeping with my promise in the introduction.
Experiments
It is tempting to guess that the needle will miss the lines as often as it crosses a line. But before we speculate, we should make an experiment. The parallel lines can be supplied by a wood floor or by a floor paved with square tiles. Lacking such a floor, one could draw parallel lines on several pieces of newsprint taped together. A piece of wire as long as the distance between the lines can serve as the needle.
When carrying out the experiment, give the needle a good spin so that it doesn't always fall at the same angle. Also, to help achieve randomness, change the direction in which you stand.
I tossed the needle 100 times. There were 66 cases in which the needle crossed a line and 34 in which it did not. That is pretty far from the guess that the 2 cases would be split evenly. The results of the first 50 of these throws are recorded in the following string of h's and m's, where an h stands for hitting a line and an m for a miss:
hhhmhmmhmhhmhmhhmhhhmhmhh mmhmhmhmhhhmmhhmhhhmhmhhh
There are 31 hits and 19 misses. Of the next 50, 35 are hits and 15 are misses.
But what are the exact odds ? I mean, as we toss the needle many times and the string of h's and m's gets longer, what will happen to the percent that are h's?
I find it strange that that percent does tend to settle down and stay closer and closer to some number. After all, the needle has no memory. Each toss is totally independent of all the earlier tosses, yet the percent of hits tends to stabilize as though the needle does remember and wants to hit a line in the long run a certain fraction of the time. We will find that fraction.
Rewording the Problem
On any toss, the needle hits either no lines or one line. To put it another way, there are either no crossings or one crossing. Thinking in terms of crossings, we may ask, "What is the average number of crossings when a needle is tossed billions of times?"
In my experiment there were 66 crossings in 100 trials. That is an average of 66/100, or 0.66 crossings per trial. Therefore, we expect our theoretical average, the one for billions of throws, to be somewhere near 0.66. Whatever the answer is for crossings, it will also tell us the likelihood of the needle's hitting a line.
The advantage of this version in terms of crossings, introduced by Émile Barbier (1839-1889) in 1870, is that it easily generalizes to other geometric shapes. We now ask, "If we have a thin wire of any shape and length, what will be the average number of crossings of the lines when we throw it billions of times?" It turns out, as we will soon see, that this more general question can easily be settled by elementary means. That the more general case turns out to be easier than the specific case is not unusual in mathematics and the sciences. The key to finding the answer may lie in asking the right question. The correct question may offer a clue to its own answer.
The Noodle
Consider any rigid wire made up of straight pieces welded together. The wire must be "flat" in the sense that when it falls on the floor, all of it touches the floor. Here are some of the possible shapes.
The wire could be straight and of any length, a letter of the alphabet, a spiral, or whatever comes to mind.
We will consider only wires in the shape of polygons. A polygon is a figure made of straight segments. In classical geometry a polygon forms a closed circuit, but we will use the term more generally.
We now ask a far more general question than the one about the needle: Handed a flat piece of wire, made of straight pieces and of a certain length and shape, we ask, "How can we predict the average number of crossings when we throw the wire many, many times?" In a 1969 paper on this question, J. F. Ramaley called this the "Buffon Noodle" problem.
For instance, the Z-shaped wire shown to the right can have 0, 1, 2, or 3 crossings. We disregard the rare case when a segment happens to lie on one of the lines. The average number of crossings must lie somewhere between 0 and 3.
This time it is hard even to guess the answer, for the average depends on the particular wire. The only case in which we have any data at this point is that of the needle, which is as long as the slats are wide.
We restrict our study to polygons — figures made up of straight-line segments — to simplify the mathematics. However, any reasonably smooth curve can be approximated by polygons made up of very short pieces, even pieces all of the same length. Because of this, our theory applies even to curves. In fact, it even applies to polygons and curves made of flexible string instead of rigid wires. For this reason we may speak of Buffon's problem for a wet noodle.
Though we seek a theoretical answer, experiments will serve as a check and may even suggest a solution. We will start with the simplest cases, a common tactic of mathematicians, almost the opposite of another tactic, illustrated in this chapter, which is to generalize.
Experiments
To describe a particular wire, we will use its length and shape. We will assume that the lines are an inch apart. That way, we can easily measure lengths with an ordinary measuring tape used in sewing.
Let us begin with a needle twice as long as the needle we started with. It is 2 inches long and thus can have 0, 1, or 2 crossings. Recall that as we throw it, we give it a good horizontal spin in order to help make the throw random.
Imagine that a bug rides the needle — a bug that will help us in later chapters as well. When the needle lands, the bug crawls from one end to the other and reports how often he crosses a line.
I threw this needle 20 times, with the number of crossings shown here in detail in the order they occurred: 1 2 2 0 2 0 2 1 2 2 2 2 1 0 1 1 2 1 2 2. There were three 0s, six 1s, and eleven 2s. So the average number of crossings per throw, that is, the total number of crossings divided by 20, is
3 x 0 + 6 x 1 + 11 x 2/20
That is, 28/20 = 1.4. On average, then, the straight wire that is 2 inches long had 1.4 crossings per throw in this experiment.
Then I bent this same wire into the shape of a V with arms of equal lengths. Here are the numbers that the bug reported for 20 throws: 2 1 2 1 2 1 2 1 1 2 1 2 1 1 2 1 1 0 1 1 . This time there were one 0, twelve 1s, and seven 2s, for a total of 26 crossings. That is an average of 1.3 crossings per throw.
Next I bent the same wire into a square. The number of crossings for the 20 throws was 0 2 2 2 0 2 2 2 2 2 2 0 2 0 0 2 2 0 0 2. (This time there could not be any 1s because when a square crosses a line, it crosses it twice. We disregard the rare case when the square just touches a line.) Since there were 26 crossings, the average per throw is 26/20 = 1.3.
Recall the 100 throws of the 1-inch-long needle, during which there were 66 crossings. In this case we see that the average was 66/100, which is 0.66 crossings per throw.
I went on to bend the same 1-inch wire into a Z. Now there could be anywhere from 0 to 3 crossings in a throw. Here are the bug's reports on 20 throws: 3 0 0 1 0 1 0 0 0 1 1 0 0 1 0 0 1 1 2 1 . This gives a total of 13 crossings in 20 throws for an average of 13/20 = 0.65 crossings per throw.
All these averages are only suggestive. They are based on only a few throws, not on thousands. However, they may guide us as we frame a theory that does not depend on any experiments.
Interpreting the Data
All our experiments are summarized in this table:
[TABLE OMITTED]
You could easily add to the list, using longer wires and other shapes. The only two factors that can influence the average are shape and length. Looking at the data, skimpy though they may be, we are tempted to say that shape has little or even no influence on the average number of crossings. For instance, changing the 2-inch wire from straight to a V and then to a square seems not to affect the average significantly. Changing the length, however, exerts a large influence. Let us take a look at the role of shape first.
The Influence of Shape
Let us use common sense to compare the two cases of a wire of length 2, straight and bent into a V. The experimental averages for these shapes were close to each other, 1.40 and 1.30.
Once again we summon bugs to assist us.
First consider the straight needle. Instead of one bug reporting crossings, let us have two bugs. Each bug wanders over his half of the needle, as shown below.
One bug reports crossings of the left section of the needle, and the other bug reports crossings of the right section. Each will report a 0 or 1 because each section of the needle is as long as the width of one slat. To learn the total number of crossings of the whole needle, we listen to the two bugs, and we add the two numbers they utter.
Neither bug knows about the other bug. Indeed, each knows only his own section and is not aware that there is another section. Each bug thinks that his section is being spun and thrown at random.
The average number of crossings for the whole needle is the sum of the averages the two bugs report. Keep this in mind as we now bend the needle into a V.
The wire, which had been straight, is now a V. The two bugs have no idea that they now ride on a bent wire. After all, each of them is aware only of his own section. Moreover, each is responsible only for reporting crossings of his section.
As the V-shaped wire is thrown at random, each of its two sections is also thrown at random. As far as the bugs are concerned, their sections are being thrown just as they were when the wire was straight. Therefore, each bug reports the number of crossings now as it did before, when it was riding half the straight needle. Therefore, the average number of crossings for each bug observed is the same as before. It follows that the (theoretical) average number of crossings for the V is the same as for the straight needle of the same length. Thus we conclude that bending the straight wire into a V of two equal arms has no effect on the average number of crossings if the wire is thrown not just 20 times but millions of times.
However, we don't know at this point what that average is. The experiments suggest that it is somewhere in the vicinity of 1.30 and 1.40.
Our analysis applies to any wire that is bent into the shape of a polygon. As a reminder, here are a few polygons, some of which we have already looked at.
To analyze the Z-shaped polygon, we call upon the service of three bugs. Place one on each of the three sides. (The sides don't have to be the same length.) Each bug has no idea that he is riding on a Z-shaped wire. He thinks he is crawling about on a much shorter straight wire and must report that section's crossings of the lines in the floor.
Now straighten the Z without dislodging the bugs, and don't even tell the bugs what you are doing. The three bugs now report on the three sections of a straight wire. They report the same number of crossings, on average, as they did when they were crawling on the Z. The figure below contrasts the "before" and "after."
When the whole Z-shaped wire is thrown at random, so is each of its three sections. The average number of crossings for the Z is therefore the same as for the straight wire of the same length. This argument applies to any polygon. Just plant a bug on each of its segments and reason as we did for three bugs. We can therefore say that shape has no influence on the average number of crossings of the wire and the lines in the floor.
The Influence of Length
Now that we have ruled out shape as an influence on the average, we are left only with length as a factor. Let us see how the average behaves as we change the length of the wire. We might as well assume that the wire is straight since that case is easiest to draw.
The straight wire of length 2 inches has an average of 1.40 crossings, which happens to be about twice the average for the wire of length 1. That comparison is based just on experiments. What do the bugs tell us about the comparison if billions of throws are made?
Imagine two bugs on the 2-inch-long wire. Each is responsible for half the wire, as shown here.
Once again each bug is unaware of the other bug and the other section. Each reports approximately the same total number of crossings when the wire is thrown billions of times. Therefore the total number of crossings for the whole wire of length 2 will tend to be twice the total number for the wire of length 1. Doubling the length will double the average number of crossings.
What if one wire is, say, 2/3 as long as another wire? In this case, divide the longer wire into three sections of equal lengths and place a bug on each section, like this.
The part of the wire, AC, is 2/3 as long as the whole wire, AB. Since there are two bugs on AC and three bugs on AB, the number of crossings of the part AC will tend to be 2/3 the number of crossings of the whole wire. This tells us that the average number of crossings for the shorter wire is 2/3 the average for the whole wire.
This type of reasoning shows that the average number of crossings for a long wire is greater than that for a short wire. Moreover, the average is proportional to the length. Another way to say this is "If, for a wire, you divide the average number of crossings by the length of the wire, you will get a number, and this number is the same for all wires."
Let us check that claim for the five wires whose data we recorded in a table:
The Missing Number
Thanks to the bugs, we know that for any wire, if we divide the average number of crossings by the length of the wire, we should get the same number as we would for any other wire. This is in fact the case for any curve or polygon. The preceding table suggests that the average divided by the length is somewhere near the range 0.65 to 0.70. But what is this missing number exactly, which is the key to our whole study of crossings? What is this "universal constant" that is not affected by shape or length?
Since the only "famous" number that is near the experimental results is 2/3, we may be tempted to guess that the missing constant is 2/3. As we will see in a moment, that guess is wrong.
Cutting more wires, bending them into various shapes, and then tossing them on the floor, even were we to throw them trillions of times, will not help us find this constant. That would give us only better estimates. Instead, to find that missing constant — the key to the whole chapter, the number that will answer Buffon's original question about a needle — we will have to figure it out by common sense, by pure thought.
(Continues...)
Excerpted from Adventures in Mathematical Reasoning by Sherman Stein. Copyright © 2001 Sherman Stein. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Table of Contents
Contents
Preface, vii,Acknowledgments, xi,
1 The Needle and the Noodle, 1,
2 Win by Two, 21,
3 The Complete Triangle, 45,
4 Slumps and Streaks, 63,
5 Thrifty Strings, 75,
6 Counting Ballots, 99,
7 Infinity, 123,
8 Twins, 137,
EPILOGUE A Backward Glance, 153,
APPENDIXES, 157,
A Triangles, 159,
B Twins: A Supplement, 163,
FOR FURTHER READING, 167,
Index, 173,