Headscratchers E-Book

Rob Eastaway

• Herausgeber: Allen & Unwin
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch
• Veröffentlichungsjahr: 2023

Seitenzahl: 195

Beschreibung

'A fantastic and varied collection of problems authored by some of the best puzzle setters around' - Alex Bellos 'A book of delightful puzzles. It's just as good as you'd expect.' - Tim Harford 'A marvellous miscellany of mysteries' - Simon Singh 'Hours of arguing and puzzling. I loved it.' - Matt Parker This highly engaging collection of 70 puzzles comes from the popular weekly column in New Scientist magazine. You'll find puzzles that are great for sharing with friends at a pub, problems drawn from real-life situations, games with intriguing strategies, and puzzles with such creative and whimsical storylines that they need to be explained to be believed. With the solutions you'll read the untold back stories behind the puzzles, and a fascinating exploration of related puzzles and mathematical ideas. You'll learn why a particular puzzle adaptation involved talking to an expert in sheep genetics, which solution was thought up by the BBC Radio 5 Drive team, and outside-the-box solutions to apparently straightforward challenges. This book is a must for any lover of puzzles or recreational mathematics.

Leseprobe

Rob Eastaway is the advisor for New Scientist’s puzzle column. He is the author of several bestselling books on everyday maths, including Maths On The Back of an Envelope, Why do Buses Come in Threes? and Maths for Mums and Dads. He lives in London.

Brian Hobbs is the creator and host of the Brain Drop Puzzles podcast and a frequent contributor to the New Scientist’s puzzle column. He lives near Dallas, Texas with his wife and five children, for whom he proudly serves as maths homework consultant.

First published in Great Britain in 2023 by Allen & Unwin, an imprint of Atlantic Books Ltd.

Copyright © Rob Eastaway, Brian Hobbs and New Scientist 2023

The moral right of Rob Eastaway, Brian Hobbs and New Scientist to be identified as the authors of this work has been asserted by them in accordance with the Copyright, Designs and Patents Act of 1988.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of both the copyright owner and the above publisher of this book.

Every effort has been made to trace or contact all copyright holders. The publishers will be pleased to make good any omissions or rectify any mistakes brought to their attention at the earliest opportunity.

10 9 8 7 6 5 4 3 2 1

A CIP catalogue record for this book is available from the British Library.

Trade paperback ISBN: 978 1 83895 877 0

E-book ISBN: 978 1 83895 878 7

Printed in Great Britain

Allen & Unwin

An imprint of Atlantic Books Ltd Ormond House

26–27 Boswell Street

London

WC1N 3JZ

www.atlantic-books.co.uk

In memory of David Singmaster, the great metagrobologist.

CONTENTS

Introduction

PART I

The Puzzles

1   Pub Puzzles

Puzzles you could do in the pub

1   Creative Addition

2   The H Coins Problem

3   The Book of Numbers

4   Late for the Gate

5   Bus Change

6   Darts Challenge

7   Evening Out

8   Caesar Cipher

9   Symmetric-L

10   Which Door?

2   Virtual(ly) Reality

Puzzles inspired by situations that really happened

11   Bone Idle

12   Sunday Drivers

13   League of Nations

14   Fastest Fingers

15   Reshuffling the Cabinet

16   Amveriric’s Boat

17   Express Coffee

18   Seventh Time Lucky?

19   The Two Ewes Day Paradox

20   The Hen Party Dorm

3   All Things Considered

Puzzles featuring everyday objects

21   Cutting the Battenberg

22   A Jigsaw Puzzle

23   The Nine Minute Egg

24   Murphy’s Law of Socks

25   Rearranging Books

26   Hidden Faces

27   Birthday Candles

28   Lightbulb Moment

29   BLOXO Cubes

30   Knight Numbers

4   Figuring It Out

Problems that need some calculation

31   Sum Thing Wrong

32   Car Crash Maths

33   Soccerdoku

34   All Squares

35   Squarebot

36   Christmas Gifts

37   Tightwad’s Safe

38   The Card Conundrum

39   Martian Food

40   Diffy

5   A Matter of Time…

Clocks, time and dates

41   Pieces of Eight

42   Which Flipping Year?

43   Six Weeks of Seconds

44   Lara’s Birthday

45   The Mountain Pass

46   A Well-Timed Nap

47   Triplet Jump

48   Half Time

49   Seeing Red

50   One of These Days

6   Mind Games

Games with intriguing strategies

51   Catch Up 5

52   Taking the Biscuit

53   A Mexican Standoff

54   My Fair Ladybug

55   Dungeons and Diagrams

56   Ant on a Tetrahedron

57   Chopping Board

58   Éclair-Voyance

59   The Goblin Game

60   Weather or Not

7   Eccentric Tales

Puzzles with peculiar characters

61   The Cake and the Candles

62   Vive la Différence

63   Diamonds are Forever

64   Changing the Guard

65   Yam Tomorrow

66   Blurri-Ness

67   No Time to Try

68   Paintings by Numbers

69   A Piazza of Dominoes

70   Trouble Brewing

PART II

Solutions, Back Stories and Commentary

PART III

Hints

Contributor Biographies

Acknowledgments

Notes

INTRODUCTION

Welcome to New Scientist’s first collection of puzzles for over 40 years. We’ve picked out 70 of our favourites from the magazine’s popular weekly column, and grouped them into themes. You’ll find puzzles that are great for sharing (or arguing over) with friends, problems drawn from real-life situations, games with intriguing strategies, and puzzles with such creative and amusing storylines that we couldn’t help but share them with you. Regardless of the category, the puzzles are designed to entertain, enlighten, and intrigue, and we hope that you enjoy them as much as we do.

A particular feature of this book is that you’ll get to peek behind the curtain and discover the previously untold backstories or concepts behind the puzzles, which are often interesting and entertaining in their own right. You’ll learn why a particular puzzle adaptation involved talking to an expert in sheep genetics, the solution that was thought up by the BBC Radio 5 Drive team, and a variety of outside-the-box solutions to seemingly simple challenges. For this reason, the ‘Solutions’ section is more substantial than in other puzzle books, and is designed to be read more thoroughly rather than simply glanced at. Indeed we expect – and maybe even hope – that some readers will find this section at least as interesting as the puzzles themselves.

It wasn’t easy to pick out only 70 of the puzzles from our collection. The list of wonderful New Scientist puzzles and the stories behind them is growing week by week. But, much like the ‘Book of Numbers’ that you’ll discover on page 8, this book has to end somewhere. The good news is that we’ll have plenty of material should we decide to produce a Volume 2.

PART 1

THEPUZZLES

Chapter 1

PUBPUZZLES

Some puzzles lend themselves to being tackled with friends. We’ll call them pub puzzles, but they work equally well in cafes, at a family lunch, on a train or in a canteen – anywhere that you can have a conversation around the table. You should be warned, however, that some of these puzzles can lead to extremely heated debate – even when you’ve read the solution.

1

CREATIVE ADDITION

ROB EASTAWAY

There is an old adage that one person’s ‘creativity’ is another person’s ‘cheating’. This puzzle will test which side of the fence you sit on.

The numbers 1 to 9 have been written on cards and left on a table:

The left-hand column adds to 21, and the right adds to 24. Your challenge is to move just one card so that the two columns add to the same total. There’s a classic ‘aha’ solution to this puzzle, but my daughter came up with a solution I wasn’t expecting. Since then I’ve been offered at least ten more distinct solutions.

How many solutions can you find that you regard as creative rather than cheating?

2

THE H COINS PROBLEM

DAVID BEDFORD

Seven coins have been placed in the ‘H’ shape below. Altogether there are five lines of three, including the diagonals.

Your challenge is to place two more coins so that you can make ten straight lines of three. No stacking of coins, and no lines of four coins or other sneaky trick is required.

If you find a way to do this, give yourself a silver medal. If you find a second way to do it that isn’t a mirror image of the first, award yourself a gold.

3

THE BOOK OF NUMBERS

HUGH HUNT

Polly plans to write a book (in English) containing all the whole numbers from zero to infinity in alphabetical order. She knows this will take her a very long time, but she makes a start. She figures that first on her alphabetical list is the number eight. After a while she tires of the task, jumps to the last page and starts working backwards. She reckons that the last entry will be zero.

Curiously, even though this book will take forever to finish writing, it’s possible to state which number will be listed second in the book, and which one will be second-last. What are those two numbers?

(Note – when Polly wants to write numbers bigger than the quadrillions, i.e. numbers with fifteen zeroes, she strings numbers together, for example ‘one billion trillion’ or ‘five million million quadrillion’.)

4

LATE FOR THE GATE

ROB EASTAWAY

This deceptively tricky everyday problem was first posed by Fields Medallist Terence Tao in 2008.

You are in a bit of a rush to catch your plane, which is leaving from a remote gate in the terminal. Some stretches of the terminal have moving walkways (travelators); other portions are carpeted.

You always walk at the same speed, but your speed is boosted when you are on the travelator.

You look down and spot that your shoelaces have come undone. This won’t slow you down, but it’s annoying, so you decide to stop to tie them. It will take the same amount of time to tie your laces if you’re on the carpet or on the travelator, but if you want to minimize the time it takes you to reach the gate, where should you tie your laces?

What if you are feeling energetic and can double your walking speed for five seconds? Is it more efficient to run while on a travelator, or on the carpet?

5

BUS CHANGE

KATIE STECKLES

I’m about to get on the bus, but the driver doesn’t give change. The fare is £1 and I don’t have exactly £1 in change on me, so I hand over more than £1 and they keep the change.

Once I’ve sat down, I realize that the amount of money I had with me was the largest possible amount I could have had in change without being able to pay £1 (or any multiple of £1) exactly. How much did I have?

(In case you need a reminder of British coinage, the coins are 1p, 2p, 5p, 10p, 20p, 50p, £1 and £2.)

6

DARTS CHALLENGE

ZOE MENSCH

On a regular dartboard the maximum that you can score with three darts is 180, by getting three treble 20s. However, there are scores below 180 that you can’t get with three darts.

What’s the lowest score you can’t get with three darts? And for that matter, what is the lowest score that you can’t get with two darts? And with one dart?

As a reminder, hitting the narrow outer ring doubles the score for that dart, and hitting the inner narrow ring triples it. Hitting the bullseye in the centre of the board scores 50 points, and hitting the ring that surrounds it scores 25.

7

EVENING OUT

PAULO FERRO

The figure above is composed of fifteen matchsticks. Move (but don’t remove!) two matchsticks to different positions to get a 3-digit number in which all three digits are even numbers. There are three solutions (none of them ‘tricks’). Can you find them all?

8

CAESAR CIPHER

ANGUS WALKER

How might Caesar get you from 3 to 47? A bit of general knowledge might help you here. Or a bit of numerology. Because surprisingly, there are two neat solutions to this puzzle.

9

SYMMETRIC-L

DONALD BELL

At first glance you wouldn’t think it’s possible to put these three L-shaped tetrominoes together to make a flat, symmetrical shape.

And yet… it turns out that there are two different ways to make a shape with mirror symmetry using all three Ls. If you can find them, you deserve an L-ympic medal.

To be clear, each L must be touching at least one other L and you are allowed to flip the Ls over, but no overlapping is allowed.

10

WHICH DOOR?

ROB EASTAWAY

You may have heard of the famous American gameshow Let’s Make a Deal, in which the star prize (a car) was hidden behind one of three doors, and the contestant had to guess which door was the lucky one.

Now there’s a new gameshow in town, Let’s Make a BIGGER Deal, hosted by Tony Macaroni. This time there are five doors instead of three, labelled A, B, C, D and E. A contestant, Kelly, is hoping to win the prize. Kelly is allowed to choose any three of the five doors. If the prize is behind one of those three doors, she wins!

Kelly picks doors A, B and D.

Now – as is always the case on the show – to build up some drama Tony Macaroni opens three doors that he knows don’t have the prize behind them. On this occasion, Macaroni opens doors A, D and E.

Two doors remain closed: one of Kelly’s choices (which is B), and door C.

Macaroni says: ‘Kelly, do you want to stick with Door B? Or do you want to switch to Door C? You can phone a friend if you want!’

Kelly likes this last suggestion and she rings her friend Jeff. ‘Hi Jeff, there are two doors left, B and C. Which one should I choose? Just give me a letter please!’

Unfortunately, Jeff gets flustered under the pressure and just blurts out a letter. What should Kelly do?

Chapter 2

VIRTUAL(LY)REALITY

Quite a few New Scientist puzzles have been inspired by situations that have cropped up in everyday life. That includes all the puzzles in this chapter, though some needed to be more heavily adapted and disguised than others. Neither of us owns a yacht in the Mediterranean, for example.

11

BONE IDLE

ROB EASTAWAY

University student Rick Sloth has spent his life avoiding work, and even though it’s exam season he still doesn’t plan to change his old habits.

He’s studying palaeontology, which he thought might be an easy option when he signed up for it, but he’s now discovered that it requires rather more study than he was expecting. It turns out there are eighteen topics in the syllabus, and his end-of-year exam will feature eleven essay questions, each on a different topic. Fortunately for Rick, in the exam the candidate is only required to answer four questions in total.

Rick wants to keep his workload to a bare minimum, while still giving himself a chance of getting full marks.

How many topics does he need to study if he is to be certain that he will have at least four questions that he will be able to tackle?

And can you come up with a general formula for how few topics you need to study based on the number of exam questions and the number of topics in the syllabus?

12

SUNDAY DRIVERS

ZOE MENSCH

The single lane road around Lake Pittoresca is stunning if you enjoy scenery, but a pain in the neck if your goal is to get to your destination fast.

Four couples staying at the Hotel Hilberto are planning a day trip to the village of Paradiso at the other end of the lake. The driver for each couple habitually takes life at a different speed. Mr Presto likes to go full throttle at every opportunity in his open-top Porsche. Mme Vivace is not quite such a speedy driver. The Andantes, meanwhile, prefer a leisurely drive, while the inconsiderate Mr and Mrs Lento creep along in second gear, ignoring any honking horns behind them.

Needless to say, if a car finds itself behind a slower car, there is no choice but to follow at the slower speed, forming a larger ‘clump’ (a clump can be formed of any number of cars, from one upwards).

On Sunday morning after breakfast all four couples set off. As luck would have it, they are the only four cars on the road. By the time they arrive at Paradiso they are in two clumps. Later, the four couples head back in reverse order, and arrive at the hotel in three clumps. Mr Presto looks particularly stressed because he didn’t have much opportunity to put his foot down on the journey back.

In which order did they set out in the morning?

13

LEAGUE OF NATIONS

ROB EASTAWAY

The TV sporting highlight of my childhood was always the Five Nations Rugby Championship, a series of matches between England, Scotland, Wales, Ireland and France. Every fortnight on Saturday afternoon there would be two matches, with the fifth country having the day off.

The fixture list had an elegant symmetry to it. Each country played every other country once, with two matches at home and two away, and each country alternated between playing at home and away.

I have a hazy memory that one year, the fixtures on the opening Saturday of the Five Nations were Ireland v England (in Dublin) and France v Wales (in Paris). On the third Saturday, Wales played at home.

If my memory is right, what were the final two matches that season?

14

FASTEST FINGERS

HOLLY BIMING

The contestants were lined up, each hoping to get into the Millionaire chair. First they would need to get through the ‘fastest fingers’ round.

The host cleared his throat:

‘List these animals in order of the number of legs they have, starting with the most:’

Guessing blindly, Jasmine went for CDBA, Virat chose CBDA, and Finnbarr picked ADCB, but none got all four right. In fact they all got the same number of answers correct in the right position.

Which has more legs, a Fettlepod or a Sentonium?

15

RESHUFFLING THE CABINET

HOLLY BIMING

The Ruritanian Prime Minister is in a bit of a fix. Thanks to a series of incompetent policy decisions, all five of her senior ministers need to be axed from their current posts. However, the PM cannot afford to sack them completely, because they’ll wreak havoc if they are relegated to the back benches.

She has a solution: a reshuffle! She will simply move each of the five ministers to one of the other top posts, but no two of them will directly swap with each other.

Anerdine will move to the department of the person who will become Chancellor. Brinkman will replace the person who will be the new Home Secretary. Crass will take over the post being vacated by the person who will take Eejit’s job. Dyer will be appointed as Health Secretary even though he’s been lobbying to become Chancellor. The current Defence Secretary will take charge of the department of the person who is becoming the Education Secretary.

Can you figure out who currently has which job, and where they are moving to?

16

AMVERIRIC’S BOAT

ROB EASTAWAY

The well-known billionaire Mr Amveriric keeps a yacht in a private dock in the Mediterranean. It is tethered to the quay by a rope.

Last time his staff tied up the boat, they left too much slack in the rope, so the boat is now one metre away from the quay when the rope is taut. Hearing that a storm is on the way, Amveriric realizes that the boat might get smashed against the wall by the buffeting wind, so he sends his henchman, Benolin Chestikov, to shorten the rope.

Seeing that the boat is one metre from the wall, Benolin decides he will pull the rope horizontally by one metre. As he pulls, the boat moves in horizontally.

The question is: will the boat reach the wall or not? (And can you prove the answer to yourself without resorting to trigonometry?)

17

EXPRESS COFFEE

DEREK COUZENS

The streets of New Addleton are set out in a rectangular grid.

Seven coffee vendors (indicated by the circles) have stalls at metro stations, and want to set up a central depot from where they can collect supplies each morning. They want to keep their combined cycling distance from stall to depot to a minimum. Pat has picked out four candidates for where to put the depot: A, B, C and D.

‘Are you sure one of those four is the optimal location?’ asks Shahin. ‘I suppose we could check out the total vendor-depot distance for every point on the grid.’

‘No need, I can confidently tell you the best place just by looking at the diagram,’ announces Kim.

Which location does Kim recommend, and why is she so confident?

18

SEVENTH TIME LUCKY?

HOLLY BIMING

Septa knows that the four digits of the PIN for her bank card are different, but apart from that her mind is a blank. She’s had six attempts so far, with no success:

Her bank has a rule of ‘seven strikes and you’re out’, so she has just one more attempt before the machine swallows her card. As it happens, she had exactly one correct digit in the right position in each of her guesses.

What’s her number?

19

THE TWO EWES DAY PARADOX

ROB EASTAWAY

Farmer Giles is thrilled that his rare-breed sheep, Lewecy, is pregnant with twins. He’s hired an expert from the genetics clinic to find out more about the lambs. The clinic has a reliable new prenatal test that looks for fragments of the lamb’s Y-chromosome circulating in Lewecy’s blood. The test has come up positive, which means that at least one of the lambs will be male.

‘I’m pleased that there’ll be another ram on the farm,’ thinks Giles, ‘but I do hope the second lamb will be female so that I’ll have two ewes on the farm next year. There’s roughly a 50-50 split between male and female lambs, so if one’s a male then it’s odds-on that the other lamb will be female.’

Is Giles right to be optimistic?

20

THE HEN PARTY DORM

ZOE MENSCH

Ten friends have rented a dormitory for the night of a hen (or bachelorette) party and each has picked a bed before heading out on the town. At 2 a.m. they get back, a little the worse for wear. Amy, the first to arrive back, can’t remember which her chosen bed was, so she just picks one at random. The next one back, Beth, heads for her own bed, but if it’s already been taken she randomly picks another. The remaining friends adopt the same approach of going to their bed if available, and randomly picking another if not.

Janice is the last to arrive back. What’s the chance that her own bed is still empty? And was Janice more or less likely to find her own bed empty than Iona, who got back just before her?

Chapter 3

ALL THINGSCONSIDERED

When was the last time you stopped and thought about your socks? Or saw the missing pieces in your jigsaw puzzle not simply as a frustration but as a frustration and an inspiration for a puzzle of a different sort? The head-scratchers in this chapter are inspired by the ordinary and everyday objects of our lives (and cake, which might not be as everyday as we’d like). These puzzles will entertain, challenge, and might just cause you to see the mundane things around you in a new light. That is, as long as you can get the lightbulb turned on.

21

CUTTING THE BATTENBERG

ANDREW JEFFREY

Lady Federica von Battenberg has baked a cake for her daughter Victoria’s birthday party. Eight children will be attending in all so eight slices are needed.

She could, of course, make seven vertical cuts to make eight identical slices. But Victoria has heard it’s possible to cut the cake into eight identical slices with only three straight cuts of the knife.

To be clear, not only must each slice be the same shape, they must all have the same amount of pink and yellow sponge and the same amount of marzipan on the outside. There’s more than one way for Lady Federica to achieve this. How many can you find?

22

A JIGSAW PUZZLE

ROB EASTAWAY