Mathematical Puzzles and Curiosities

By Barry R. Clarke

"Very satisfying." — Will Shortz, Crossword Editor, The New York Times. This new collection features an intriguing mix of recreational math, logic, and creativity puzzles, many of which first appeared in the author's Daily Telegraph (UK) column. Requiring only basic algebra skills, classic and new puzzles include The Monty Hall Problem, The Unexpected Hanging, The Shakespeare Puzzles, and Finger Multiplication.

• Language: English
• Publisher: Dover Publications
• Release date: Jun 19, 2013
• ISBN: 9780486315720

Book preview

A thoroughly enjoyable and mind-expanding array of puzzles and curiosities

Dr Cliff Pickover, author of Archimedes to Hawking

Clever, original brain-teasers are rare. This book has some beauties. Very satisfying.

Will Shortz, Crossword Editor, The New York Times

I highly recommend this delightful book. It contains not only excellent puzzles, but also extremely interesting commentaries and anecdotes.

Professor Raymond Smullyan, author of To Mock a Mockingbird

A wide-ranging and attractive collection that will appeal to all puzzle fans

Professor Ian Stewart, author of Professor Stewart’s Cabinet of Mathematical Curiosities

A masterpiece ... reminds me somewhat of my first introduction to Martin Gardner ... this is a must for all puzzle lovers worldwide

Terry Stickels, author of Frame Games

MATHEMATICAL

PUZZLES

&

CURIOSITIES

Barry R. Clarke

DOVER PUBLICATIONS. INC

Mineola, New York

Acknowledgments

I should like to thank Denis Borris and Mark Rickert for testing some of the mathematics and logic puzzles in this work on the puzzles forum of my website http://barryispuzzled.com. Also, thanks to my many mathematics students for trying out some of the creative thinking puzzles and providing valuable feedback. I am grateful to Val Gilbert, and Alex and Kate Ware, for providing the opportunity to construct some of these puzzles for The Daily Telegraph. Finally, I am grateful to Rochelle Kronzek and James Miller from Dover Publications for their commitment to this work .

Copyright

Copyright © 2013 by Barry R. Clarke

All rights reserved.

Bibliographical Note

Mathematical Puzzles and Curiosities is a new work, first published by Dover Publications, Inc., in 2013.

International Standard Book Number

eISBN-13: 978-0-486-31572-0

Manufactured in the United States by Courier Corporation

49091201 2013

www.doverpublications.com

Contents

Introduction

The Monty Hall Problem

Mathematical Puzzles I

1.The Baffled Brewer

2.Horse Play

3.Court Out

4.Back to Class

5.Sound Arithmetic

6.Digital Dilemma.

7.Core Conundrum

8.Word in the Stone

9.The Backward Robber

10.Maximum Security

Judgment Paradoxes

Finger Multiplication

Creative Thinking Puzzles I

11.Mad House

12.Right Angle

13.Amazing

14.The Dead Dog.

15.Sum Line

16.Which Way?

17.Nothing for It

18.Rough Graph

19.The Lighthouse

20.The Pig and the Bird

The Sleeping Beauty Problem

Logic Puzzles

21.Shilly Chalet.

22.The Five Chimneys

23.A Pressing Problem

24.Alien Mutations

25.Santa Flaws

26.Safe Cracker

27.A Raft of Changes.

Parity Tricks with Coins

The Shakespeare Puzzles.

Creative Thinking Puzzles II

28.Inspector Lewis

29. The Builder’s Problem

30.The Four Dice

31.Politically Correct

32.Inklined

33.Bubble Math.

34.Arithmystic

35.No Escape

36.Secret City

37.Winning Line

The Hardest Ever Logic Puzzle

Zeno and Infinitesimals

Mathematical Puzzles II

38.The Striking Clock

39.Square Feet with Corn.

40.The Broken Ruler

41.Play on Words

42.Fare Enough

43.Having a Ball

44.Oliver’s Digeridoo

45.Neddy’s Workload

46.Armless Aliens.

47. The Slug and the Snail

The Unexpected Hanging

The Wave-Particle Puzzle

Creative Thinking Puzzles III

48.Moving Clocks

49.Miserable Marriage

50.Out of this World

51.Doubtful Date

52.Door to Door

53.No Earthly Connection

54.Water Puzzle.

55.Doing a Turn

56.The Tin Door

57.Fish Feast

58.Dig It

59.The Concealed Car

60.Pet Theory

Titan’s Triangle.

Creative Thinking Hints

Solutions

Introduction

First, here’s a little enigma which at first sight seems trivial, but if you keep strictly to the given condition for solving it, then it’s not so easy. Below are seven letters that form an anagram — and a seven-letter anagram is reasonably straightforward — but the puzzle is, how can you reach the solution without rearranging the letters? The answer, given at the end of the Introduction, demands a leap of the imagination but remember, if you succumb to the temptation to move the letters around then you’ve cheated!

So, welcome, and I hope that you enjoy this original collection of puzzles and articles. It is a carefully considered compilation in which you can find conundrums in logic, mathematics, and creativity, together with some thought-provoking articles in recreational mathematics and philosophy. It is ideal for those who enjoy alternative ways of thinking and who like to consider a fresh approach to problems. Most of the book requires no specialised mathematical knowledge but those articles near the end that are more demanding can be penetrated by a reasonable amount of high-school algebra.

Some of the articles deal with classic teasers such as The Unexpected Hanging, The Monty Hall Problem, and The Sleeping Beauty Problem, but I have resisted resurrecting a standard analysis of these items, preferring instead to present my own way of understanding them. Other topics such as the Shakespeare Puzzles and Titan’s Triangle are entirely new, the first being a creative interpretation of the dedications that preface the Shakespeare Sonnets (1609) and First Folio (1623), and the latter being a fascinating extension of a classic IQ puzzle. There are also more philosophical topics such as Zeno and Infinitesimals, and the Wave-Particle Puzzle, which I hope will encourage the reader to think again about these problems.

The puzzles in this work are entirely original and most of them have been published in my column in The Daily Telegraph. They have been arranged to increase in difficulty as the pages turn and the solutions have been deliberately placed out of order at the end to avoid inadvertently seeing the next solution. The answer to a puzzle can be located by referring to the solution number given at the end of the puzzle (not the page number) then looking it up at the back of the book. The creative thinking puzzles encourage alternative ways of thinking and two hints for each are provided near the end of the book to lessen the demand for mind reading. They are ideal for group problem-solving sessions and many of them have already been tested on students who have found them engaging, stimulating, and often amusing. If the solution remains beyond your grasp, even after pondering the hints, please don’t feel frustrated. My wish for you is that on examining the answer you can enjoy it as puzzle art and, perhaps after having mastered a few of the basic principles, might even be inspired to create some of your own.

The following is an example of the kind of visual creative thinking puzzle that might be encountered within these covers. The solution is given at the end of the Introduction. Can you spot the difference between these two quarters?

Having taught mathematics at various independent sixth-form colleges in Oxford for many years, one of my interests is in the role that puzzles, especially creative thinking ones, can play in developing the young mind. Sadly, the sole aim of our education system as it currently stands is entirely materialistic, this being to prepare students to obtain employment and earn a living. This usually requires qualifications such as a school certificate or university degree and our current school education system is one part of the long conveyor belt that serves this end. Of course, there are bills to pay so earning a living is far from being undesirable, but my point is that if this is the only goal then there is a price to pay.

When the teacher of mathematics presents his material, he does so with the intention of enabling his students to pass examinations. Mathematics examinations consist of questions and each question demands one or more methods to successfully negotiate it. If the student can identify the set of techniques required for each particular problem and accurately apply them then he can obtain a good result in the examination. The inquisitive student might express reservations about the usefulness of a particular problem to his future existence, ask for an insight into the history of the development of a certain topic, or even request a deeper understanding of why a method works, but none of this is vital to achieving the eventual goal. To pass mathematics examinations, it is only necessary to understand the how and not the why. A sympathetic teacher might wish to deviate from his rote-learning program and nurture the student’s own capacity for independent thought, but with little time available to cover the curriculum it only compromises his own ability to deliver examination-successful students. Put another way, the current school system does not stimulate the student to think independently, and since most mathematics teachers enjoy exploring the background to their subject, I believe that given the choice they would prefer their students to do likewise.

This exclusive focus on examination results is a pity as far as the development of the student is concerned because there are a number of not only intellectual but also social advantages to be had by encouraging the student to explore and be creative. In the first place, a student who is engaged in creative activity is trying to invent possible ways of tackling a problem. This ability to think flexibly, which needs to be developed through repeated use, is vital to socially adaptive behaviour and leads to better social coping mechanisms. Also, a student who is trying out various possibilities to solve a problem, is bound to fail with some of the alternatives. So long as his educational environment is non-judgmental, it is possible for him to learn that this failure is a necessary part of problem solving, that it has a positive value in discovering what to eliminate from the enquiry, and that the next step is to try out other possibilities rather than fall into morbid introspection. The ability to take a risk in life, to fail, and then to be ready to try out a different approach, is a highly desirable skill.

It is unfortunate that the first real opportunity the student gets to truly explore original ideas usually only presents itself as late as doctorate level, by which time the young mind has been so deeply engraved with unquestionable ‘truths’ that it is almost impossible to think beyond them. Even then, some scientific PhD programs are so constrained they are nothing more than a passable permutation of the research department’s existing ideas, accepted for publication in journals by devotees to the established order. There is nothing new in this circumstance, for it has existed since the birth of scientific enquiry in the