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Overview
"Does your brain need a workout? Are you wondering what to do? Well this could be the book for you! This is a great book for anyone who is looking for a challenge and, if you get too stuck, the author kindly also provided the answers at the end of each chapter!" — Splashes Into Books
Give your mind a playful workout with this collection of more than 100 inventive puzzles. Finding the solutions requires only minimal mathematical knowledge and will test your imagination as well as your brainpower. The motley collection ranges from conundrums and mathematical stunts to practical situations involving counting and measuring. Chess problems, magic squares, and a host of other intriguing scenarios will amuse and challenge puzzle enthusiasts and fans of recreational mathematics. Answers appear at the end of each chapter.
These puzzles are the inventions of a gifted Soviet mathematician, Yakov Perelman, whose popular science books on astronomy, physics, and mathematics inspired generations of readers. Perelman's distinctive style, abounding in wit and ingenuity, adds a special flair to his timeless riddles and brainteasers.
Product Details
| ISBN-13: | 9780486803456 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 02/02/2015 |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 144 |
| File size: | 8 MB |
About the Author
Read an Excerpt
Figures For Fun
Stories, Puzzles and Conundrums
By Yakov Perelman
Dover Publications, Inc.
Copyright © 2015 Dover Publications, Inc.All rights reserved.
ISBN: 978-0-486-80345-6
CHAPTER 1
BRAIN-TEASERS FOR LUNCH
It was raining.... We had just sat down for lunch at our holiday home when one of the guests asked us whether we would like to hear what had happened to him in the morning.
Everyone assented, and he began.
1. A SQUIRREL IN THE GLADE.— "I had quite a bit of fun playing hide-and-seek with a squirrel," he said. "You know that little round glade with a lone birch in the centre? It was on this tree that a squirrel was hiding from me. As I emerged from a thicket, I saw its snout and two bright little eyes peeping from behind the trunk. I wanted to see the little animal, so I started circling round along the edge of the glade, mindful of keeping the distance in order not to scare it. I did four rounds, but the little cheat kept backing away from me, eyeing me suspiciously from behind the tree. Try as I did, I just could not see its back."
"But you have just said yourself that you circled round the tree four times," one of the listeners interjected.
"Round the tree, yes, but not round the squirrel."
"But the squirrel was on the tree, wasn't it?"
"So it was."
"Well, that means you circled round the squirrel too."
"Call that circling round the squirrel when I didn't see its back?"
"What has its back to do with the whole thing? The squirrel was on the tree in the centre of the glade and you circled round the tree. In other words, you circled round the squirrel."
"Oh no, I didn't. Let us assume that I'm circling round you and you keep turning, showing me just your face. Gall that circling round you?"
"Of course, what else can you call it?"
"You mean I'm circling round you though I'm never behind you and never see your back?"
"Forget the back! You're circling round me and that's what counts. What has the back to do with it?"
"Wait. Tell me, what's circling round anything? The way I understand it, it's moving in such a manner so as to see the object I'm moving around from all sides. Am I right, professor?" He turned to an old man at our table.
"Your whole argument is essentially one about a word," the professor replied. "What you should do first is agree on the definition of 'circling.' How do you understand the words 'circle round an object'? There are two ways of understanding that. Firstly, it's moving round an object that is in the centre of a circle. Secondly, it's moving round an object in such a way as to see all its sides. If you insist on the first meaning, then you walked round the squirrel four times. If it's the second that you hold to, then you did not walk round it at all. There's really no ground for an argument here, that is, if you two speak the same language and understand words in the same way."
"All right, I grant there are two meanings. But which is the correct one?"
"That's not the way to put the question. You can agree about anything. The question is, which of the two meanings is the more generally accepted? In my opinion, it's the first and here's why. The sun, as you know, does a complete circuit in 26 days...."
"Does the sun revolve?"
"Of course, it does, like the earth. Just imagine, for instance, that it would take not 26 days, but 365¼ days, i.e., a whole year, to do so. If this were the case, the earth would see only one side of the sun, that is, only its 'face.' And yet, can anyone claim that the earth does not revolve round the sun?"
"Yes, now it's clear that I circled round the squirrel after all."
"I've a suggestion, comrades!" one of the company shouted. "It's raining now, no one is going out, so let's play riddles. The squirrel riddle was a good beginning. Let each think of some brain-teaser. "
"I give up if they have anything to do with algebra or geometry," a young woman said.
"Me too," another joined in.
"No, we must all play; but we'll promise to refrain from any algebraical or geometrical formulas, except, perhaps, the most elementary ones. Any objections?"
"None!" the others chorussed. "Let's go."
"One more thing. Let professor be our judge."
2. SCHOOL-GROUPS.— "We have five extra-curricular groups at school," a Young Pioneer began. "They're political, literary, photographic, chess and choral groups. The political group meets every other day, the literary every third day, the photographic every fourth day, the chess every fifth day and the choral every sixth day. These five groups first met on January 1 and thenceforth meetings were held according to schedule. The question is, how many times did all the five meet on the one and same day in the first quarter (January 1 excluded)?"
"Was it a Leap Year?"
"No."
"In other words, there were 90 days in the first quarter."
"Right."
"Let me add another question," the professor broke in. "It's this: how many days were there when none of the groups met in that first quarter?"
"So, there's a catch to it? There'll be no other evening when all the five groups meet and no evening when some do not meet. That's clear!"
"Why?"
"Don't know. But I've a feeling there's a catch."
"Comrades!" said the man who had suggested the game. "We won't reveal the results now. Let's have more time to think about them. Professor will announce the answers at supper."
3. WHO COUNTED MORE?– "Two persons, one standing at the door of his house and the other walking up and down the pavement, were counting passers-by for a whole hour. Who counted more?"
"Naturally the one walking up and down," said somebody at the end of the table.
"We'll know the answer at supper," the professor said. "Next!"
4. GRANDFATHER AND GRANDSON.— "In 1932 I was as old as the last two digits of my birth year. When I mentioned this interesting coincidence to my grandfather, he surprised me by saying that the same applied to him too. I thought that impossible...."
"Of course that's impossible," a young woman said.
"Believe me, it's quite possible and grandfather proved it too. How old was each of us in 1932 ?"
5. RAILWAY TICKETS. — "I'm a railway ticket seller," said the next person, a young lady. "People think this job is easy. They probably have no idea how many tickets one has to sell, even at a small station. There are 25 stations on my line and different tickets for each section up and down the line. How many different kinds of tickets do you think I have at my station?"
"Your turn next," the professor said to a flier.
6. A DIRIGIBLE'S FLIGHT.— "A dirigible took off from Leningrad in a northerly direction. Five hundred kilometres away it turned and flew 500 kilometres eastward. After that it turned south and covered another 500 kilometres. Then it flew 500 kilometres in a westerly direction, and landed. The question is, where did it land: west, east, north or south of Leningrad?"
"That's an easy one," someone said. "Five hundred steps forward, 500 to the right, 500 back and 500 to the left, and you're naturally back where you'd started from!"
"Easy? Well then, where did the dirigible land?"
"In Leningrad, of course. Where else?"
"Wrong!"
"Then I don't understand."
"Yes, there's some catch to this puzzle," another joined in. "Didn't the dirigible land in Leningrad?"
"Won't you repeat your problem?"
The flier did. The listeners looked at each other.
"All right," the professor said. "We have enough time to think about the answer. Let's have the next one now."
7. SHADOW. — "My puzzle," said the next man, "is also about a dirigible. What's longer, the dirigible or its perfect shadow?"
"Is that all?"
"It is."
"Well, then. The shadow is naturally longer than the dirigible: sunrays spread fanlike, don't they?"
"I wouldn't say so," another interjected. "Sunrays are parallel to each other and that being so, the dirigible and its shadow are of the same size."
"No, they aren't. Have you ever seen rays spreading from behind a cloud? If you have, you've probably noticed how much they spread.
The shadow of the dirigible must be considerably bigger than the dirigible itself, just as the shadow of a cloud is bigger than the cloud itself."
"Then why is it that people say that sunrays are parallel to each other? Seamen, astronomers, for instance."
The professor put a stop to the argument by asking the next person to go ahead with his conundrum.
8. MATCHES. —The man emptied a box of matches on the table and divided them into three heaps.
"You aren't going to start a bonfire, are you?" someone quipped.
"No, they're for my brain-teaser. Here you are—three uneven heaps. There are altogether 48 matches. I won't tell how many there are in each heap. Look well. If I take as many matches from the first heap as there are in the second and add them to the second, and then take as many from the second as there are in the third, and add them to the third, and finally take as many from the third as there are in the first and add them to the first—well, if I do all this, the heaps will all have the same number of matches. How many were there originally in each heap?"
9. THE "WONDERFUL" STUMP.— "My puzzle is the one I was once asked by a village mathematician to solve," the next person began. "It was really a story, and quite humorous at that. One day, a peasant met an old man in a forest. The two fell into a conversation and the latter said:
"'There's a wonderful little stump in this forest. It helps people in need."
"'It does? What does it do, cure people?'
"'Not exactly. It doubles one's money. Put your pouch among the roots, count one hundred and—presto!—the money's doubled. It's a wonderful stump, that!'
"'Can I try it?' the peasant asked excitedly.
"'Why not? Only you must pay.'
"'Pay whom and how much?'
"'The man who shows you the stump. That's me. As to how much, that's another matter.'
"The two men began to bargain. When the old man learned that the peasant did not have much money, he agreed to take 1 ruble 20 kopeks every time the money doubled.
"The two went deep into the forest where, after a long search, the old man brought the peasant to a moss-covered fir stump in bushes. He then took the peasant's pouch and shoved it among the roots. After that they counted one hundred. The old man took a long time to find the pouch and returned it to the peasant,
"The latter opened the pouch and, lo! The money really had doubled! He counted off the ruble and 20 kopeks, as agreed upon, and asked the old man to repeat the whole thing.
"Once again they counted one hundred, once again the old man began his search for the pouch and once again there was a miracle— the money had doubled again. And just as they had agreed, the old man got his ruble and 20 kopeks.
"Then they hid the pouch for the third time and this time too the money doubled. But after the peasant had paid the old man his ruble and 20 kopeks, there was nothing left in the pouch. The poor fellow had lost all his money in the process. There was no more money to be doubled and he walked off crest-fallen.
"The secret, of course, is clear to all—it was not for nothing that the old man took so long to find the pouch. But there is another question I would like to ask you: how much did the peasant have originally?"
10. THE DECEMBER PUZZLE.— "Well, comrades," began the next man. "I'm a linguist, not a mathematician, so you needn't expect a mathematical problem. I'll ask you one of another kind, one close to my sphere of activity. It's about the calendar."
"Go ahead."
"December is the twelfth month of the year. Do you know what the name really means? The word comes from the Greek 'deka'—ten. Hence, decalitre which means ten litres, decade—ten years, etc. December, to all appearances, should be the tenth month and yet it isn't. How d'you explain that?"
11. AN ARITHMETICAL TRICK.— "I'll give you an arithmetical trick and ask you to explain it. One of you—you, professor, if you like—write down a three-digit number, but don't tell me what it is."
"Can we have any noughts in it?"
"I set no reservations. You can write down any three numerals you want."
"All right, I've done it. What next?"
"Write tie same number alongside. Now you have a six-digit number."
"Right."
"Pass the slip to your neighbour, the one farther away from me, and let him divide this six-digit number by seven."
"It's easy for you to say that, and what if it can't be done?"
"Don't worry, it can."
"How can you be so sure when you haven't seen the number?"
"We'll talk after you've divided it."
"You're right. It does divide."
"Now pass the result to your neighbour, but don't tell me what it is.
Let him divide it by 11."
"Think you'll have your own way again?"
"Go ahead, divide it. There'll be no remainder."
"You're right again. Now what?"
"Pass the result on and let the next man divide it, say, by 13."
"That's a bad choice. There are very few numbers that are divisible by 13.... You're certainly lucky, this one is!"
"Now give me the slip, but fold it so that I don't see the number."
Without unfolding the slip, the man passed it on to the professor.
"Here's your number. Correct?"
"Absolutely." The professor was surprised. "That is the number I wrote down.... Well, everyone has had his turn, the rain has stopped, so let's go out. We'll know the answers tonight. You may give me all the slips now."
Answers 1 to 11
1. The squirrel puzzle was explained earlier, so we'll pass on to the next.
2. We can easily answer the first question: how many times did all the five groups meet on one and the same day in the first quarter (January 1 excluded) by finding the least common multiple of 2, 3, 4, 5 and 6. That isn't difficult. It's 60. Therefore, the five will all meet again on the 61st day—the political group after 30 two-day intervals, the literary after 20 three-day intervals, the photographic after 15 fourday intervals, the chess after 12 five-day intervals and the choral after 10 six-day intervals. In other words, they can meet on the one and same day only once in 60 days. And since there are 90 days in the first quarter, it means there can be only one other day on which they all meet.
It is much more difficult to find the answer to the second question: how many days are there when none of the groups meets in the first quarter? To find that, it is necessary to write down all the numbers from 1 to 90 and then cross out all the days when the political group meets: e.g., 1, 3, 5, 7, 9, etc. After that one must cross out the literary group days: e.g., 4, 7, 10, etc. When all the photographic, chess and choral groups' days have also been crossed out, the numbers that remain are the days when there is no group meeting.
Do that and you'll see that there are 24 such days—eight in January, i.e., 2, 8, 12, 14, 18, 20, 24 and 30, seven in February and nine in March.
3. Both of them counted the same number of passers-by. While the one who stood at the door counted all those who passed both ways, the one who was walking counted all the people he met going up and down the pavement.
4. At first it may seem that the problem is incorrectly worded, that both grandfather and grandson are of the same age. We shall soon see that there is nothing wrong with the problem.
It is obvious that the grandson was born in the 20th century. Therefore, the first two digits of his birth year are 19 (the number of hundreds). The other two digits added to themselves equal 32. The number therefore is 16: the grandson was born in 1916 and in 1932 he was 16.
The grandfather, naturally, was born in the 19th century. Therefore, the first two digits of his birth year are 18. The remaining digits multiplied by 2 must equal 132. The number sought is half of 132, i.e., 66. The grandfather was born in 1866 and in 1932 he was 66.
Thus, in 1932 the grandson and the grandfather were each as old as the last two digits of their birth years.
5. At each of the 25 stations passengers can get tickets for any of the other 24. Therefore, the number of different tickets required is: 25×24=600.
6. There is nothing contradictory in this problem. The dirigible did not fly along the contours of a square. It should be borne in mind that the earth is round and that the meridians converge at the poles (fig. 2). Therefore, flying 500 kilometres along the parallel 500 kilometres north of Leningrad latitude, the dirigible covered more degrees going eastward than it did when it was returning along Leningrad latitude. As a result, the dirigible completed its flight east of Leningrad.
How many kilometres away? That can be calculated. Fig. 2 shows the route taken by the dirigible: ABCDE. N is the North Pole where meridians AB and DC meet. The dirigible first flew 500 kilometres northward, i.e., along meridian AN. Since the degree of a meridian is 111 kilometres long, the 500-kilometre-long arc of the meridian is equal to 500:111 ==4°5'. Leningrad lies on the 60th parallel. B, therefore, is on 60° +4°5'=64°5'. The airship then flew eastward, i.e., along the BG parallel, covering 500 kilometres. The length of one degree of this parallel may be calculated (or learned from tables); it is equal to 48 kilometres. Therefore, it is easy to determine how many degrees the dirigible covered in its eastward flight: 500:48=10°4'. Continuing, the airship flew southward, i.e., along meridian CD, and, having covered 500 kilometres, returned to the Leningrad parallel. Thence the way lay westward, i.e., along DA; the 500 kilometres of this way are obviously less than the distance between A and D. There are as many degrees in AD as in BC, i.e., 10°4'. But the length of 1° at the 60th parallel equals 55.5 kilometres. Therefore, the distance between A and D is equal to 55.5×10.4=577 kilometres. We see thus that the dirigible could not have very well landed in Leningrad: it landed 77 kilometres away, on Lake Ladoga.
(Continues...)
Excerpted from Figures For Fun by Yakov Perelman. Copyright © 2015 Dover Publications, Inc.. 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,CHAPTER I Brain-Teasers for Lunch,
CHAPTER II Mathematics in Games,
CHAPTER III Another Dozen Puzzlers,
CHAPTER IV Counting,
CHAPTER V Bafflers With Numbers,
CHAPTER VI Number Giants,
CHAPTER VII Without Instruments of Measurement,
CHAPTER VIII Geometric Brain-Teasers,
CHAPTER IX The Geometry of Rain and Snow,
CHAPTER X Mathematics and the Deluge,
CHAPTER XI Thirty Different Problems,
Answers 91 to 120,