REVISED EDITION DK LONDON Project editor Vicky Richards Art editor Chrissy Barnard Editor Rona Skene Managing editor Francesca Baines Managing art editor Philip Letsu Production editor Kavita Varma Production controller Samantha Cross Jacket design development manager Sophia MTT Publisher Andrew Macintyre Associate publishing director Liz Wheeler Art director Karen Self Publishing director Jonathan Metcalf DK DELHI Senior art editor Mahua Sharma Editor Upamanyu Das MATHs Managing editor Kingshuk Ghoshal Managing art editor Sudakshina Basu DTP designers Bimlesh Tiwary, Rakesh Kumar Jacket designer Tanya Mehrotra FIRST EDITION Senior editor Francesca Baines Project editors Clare Hibbert, James Mitchem Designer Hoa Luc Senior art editors Jim Green, Stefan Podhorodecki Additional designers Dave Ball, Jeongeun Yule Park Managing editor Linda Esposito Managing art editor Diane Peyton Jones Category publisher Laura Buller Production editor Victoria Khroundina Senior production controller Louise Minihane Jacket editor Manisha Majithia Jacket designer Laura Brim Picture researcher Nic Dean DK picture librarian Romaine Werblow Publishing director Jonathan Metcalf Associate publishing director Liz Wheeler Art director Phil Ormerod This edition published in 2022 First published in Great Britain in 2012 by Dorling Kindersley Limited, DK, One Embassy Gardens, 8 Viaduct Gardens, London, SW11 7BW The authorised representative in the EEA is Dorling Kindersley Verlag GmbH. Arnulfstr. 124, 80636 Munich, Germany TThhiiss bbooookk iiss ffuullll ooff ppuuzzzzlleess aanndd aaccttiivviittiieess ttoo bboooosstt yyoouurr bbrraaiinn Copyright © 2012, 2022 Dorling Kindersley Limited A Penguin Random House Company ppoowweerr.. TThhee aaccttiivviittiieess aarree aa lloott ooff 10 9 8 7 6 5 4 3 2 1 ffuunn,, bbuutt yyoouu sshhoouulldd aallwwaayyss cchheecckk 001– 324978–Jan/2022 wwiitthh aann aadduulltt bbeeffoorree yyoouu ddoo aannyy All rights reserved. No part of this publication may ooff tthheemm ssoo tthhaatt tthheeyy kknnooww wwhhaatt be reproduced, stored in or introduced into a retrieval yyoouu aarree ddooiinngg aanndd aarree ssuurree system, or transmitted, in any form, or by any means (electronic, tthhaatt yyoouu aarree ssaaffee.. mechanical, photocopying, recording, or otherwise), without the prior written permission of the copyright owner. A CIP catalogue record for this book is available from the British Library. ISBN: 978-0-2415-1524-2 Printed and bound in China www.dk.com This book was made with Forest Stewardship Council ™ certified paper – one small step in DK’s commitment to a sustainable future. For more information go to www.dk.com/our-green-pledge 000022__000033__118822443388__IImmpprriinntt__ttiittllee..iinndddd 22 0022//0099//2211 66::3399 PPMM REVISED EDITION DK LONDON Project editor Vicky Richards Art editor Chrissy Barnard Editor Rona Skene Managing editor Francesca Baines Managing art editor Philip Letsu Production editor Kavita Varma Production controller Samantha Cross Jacket design development manager Sophia MTT HOW TO BE A Publisher Andrew Macintyre Associate publishing director Liz Wheeler Art director Karen Self Publishing director Jonathan Metcalf DK DELHI Senior art editor Mahua Sharma Editor Upamanyu Das MMAATTHHss Managing editor Kingshuk Ghoshal Managing art editor Sudakshina Basu DTP designers Bimlesh Tiwary, Rakesh Kumar Jacket designer Tanya Mehrotra FIRST EDITION Senior editor Francesca Baines Project editors Clare Hibbert, James Mitchem Designer Hoa Luc Senior art editors Jim Green, Stefan Podhorodecki Additional designers Dave Ball, Jeongeun Yule Park Managing editor Linda Esposito Managing art editor Diane Peyton Jones Category publisher Laura Buller Production editor Victoria Khroundina Senior production controller Louise Minihane Jacket editor Manisha Majithia Jacket designer Laura Brim Picture researcher Nic Dean GENIUS DK picture librarian Romaine Werblow Publishing director Jonathan Metcalf Associate publishing director Liz Wheeler Art director Phil Ormerod This edition published in 2022 First published in Great Britain in 2012 by Dorling Kindersley Limited, DK, One Embassy Gardens, 8 Viaduct Gardens, London, SW11 7BW The authorised representative in the EEA is Dorling Kindersley Verlag GmbH. Arnulfstr. 124, 80636 Munich, Germany TThhiiss bbooookk iiss ffuullll ooff ppuuzzzzlleess aanndd aaccttiivviittiieess ttoo bboooosstt yyoouurr bbrraaiinn Copyright © 2012, 2022 Dorling Kindersley Limited A Penguin Random House Company ppoowweerr.. TThhee aaccttiivviittiieess aarree aa lloott ooff 10 9 8 7 6 5 4 3 2 1 ffuunn,, bbuutt yyoouu sshhoouulldd aallwwaayyss cchheecckk 001– 324978–Jan/2022 wwiitthh aann aadduulltt bbeeffoorree yyoouu ddoo aannyy All rights reserved. No part of this publication may ooff tthheemm ssoo tthhaatt tthheeyy kknnooww wwhhaatt be reproduced, stored in or introduced into a retrieval yyoouu aarree ddooiinngg aanndd aarree ssuurree system, or transmitted, in any form, or by any means (electronic, tthhaatt yyoouu aarree ssaaffee.. mechanical, photocopying, recording, or otherwise), without the prior written permission of the copyright owner. A CIP catalogue record for this book is available from the British Library. ISBN: 978-0-2415-1524-2 Printed and bound in China www.dk.com Written by Dr Mike Goldsmith This book was made with Forest Consultant Branka Surla Stewardship Council ™ certified paper – one small step in DK’s commitment to a sustainable future. For more information go to Illustrated by Seb Burnett www.dk.com/our-green-pledge 000022__000033__118822443388__IImmpprriinntt__ttiittllee..iinndddd 22 0022//0099//2211 66::3399 PPMM 000022__000033__118822443388__IImmpprriinntt__ttiittllee..iinndddd 33 0022//0099//2211 66::3399 PPMM CCOONNTTEENNTTSS 6 A world of maths MMAATTHHSS BBRRAAIINN IINNVVEENNTTIINNGG NNUUMMBBEERRSS MMAAGGIICC NNUUMMBBEERRSS 26 Learning to count 10 Meet your brain 50 Seeing sequences 28 Number systems 12 Maths skills 52 Pascal’s triangle 30 Big zero 14 Learning maths 54 Magic squares 32 Pythagoras 16 Brain vs. machine 56 Missing numbers 34 Thinking outside the box 18 Problems with numbers 58 Karl Gauss 36 Number patterns 20 Early mathematicians 60 Infinity 38 Calculation tips 22 Seeing the solution 62 Numbers with meaning 40 Archimedes 64 Number tricks 42 Maths and measures 66 Puzzling primes 44 How big? How far? 46 The size of the problem 4 000044__000055__118822443388__CCOONNTTEENNTTSS..iinndddd 44 2277//0088//2211 66::0099 PPMM CCOONNTTEENNTTSS 6 A world of maths MMAATTHHSS BBRRAAIINN IINNVVEENNTTIINNGG NNUUMMBBEERRSS MMAAGGIICC NNUUMMBBEERRSS SSHHAAPPEESS AANNDD SSPPAACCEE AA WWOORRLLDD OOFF MMAATTHHSS 10 Meet your brain 26 Learning to count 50 Seeing sequences 70 Triangles 94 Interesting times 120 Glossary 12 Maths skills 28 Number systems 52 Pascal’s triangle 72 Shaping up 96 Mapping 122 Answers 14 Learning maths 30 Big zero 54 Magic squares 74 Shape shifting 98 Grace Hopper 126 Index 16 Brain vs. machine 32 Pythagoras 56 Missing numbers 76 Round and round 100 Probability 128 Acknowledgments 18 Problems with numbers 34 Thinking outside the box 58 Karl Gauss 78 The third dimension 102 Displaying data 20 Early mathematicians 36 Number patterns 60 Infinity 80 3-D shape puzzles 104 Logic puzzles and paradoxes 22 Seeing the solution 38 Calculation tips 62 Numbers with meaning 82 3-D fun 106 Breaking codes 40 Archimedes 64 Number tricks 84 Leonhard Euler 108 Codes and ciphers 42 Maths and measures 66 Puzzling primes 86 Amazing mazes 110 Alan Turing 44 How big? How far? 88 Optical illusions 112 Algebra 46 The size of the problem 90 Impossible shapes 114 Brainteasers 116 Secrets of the Universe TThhee bbooookk iiss ffuullll ooff pprroobblleemmss aanndd ppuuzzzzlleess ffoorr 118 Katherine Johnson yyoouu ttoo ssoollvvee.. TToo cchheecckk tthhee aannsswweerrss ttuurrnn ttoo ppaaggeess 112222––112255.. 4 5 000044__000055__118822443388__CCOONNTTEENNTTSS..iinndddd 44 2277//0088//2211 66::0099 PPMM 000044__000055__118822443388__CCOONNTTEENNTTSS..iinndddd 55 2277//0088//2211 66::0099 PPMM TThheerree’’ss aa hheeiigghhtt rreessttrriiccttiioonn oonn tthhiiss rriiddee,, ssoonnnnyy.. TTrryy II´´llll bbee iinn tthhiiss qquueeuuee ffoorr ccoommiinngg bbaacckk nneexxtt yyeeaarr.. 1100 mmiinnuutteess,, ssoo II sshhoouulldd ssttiillll bbee iinn ttiimmee ttoo ccaattcchh tthhee nneexxtt II wwoonnddeerr wwhhaatt wwoouulldd bbuuss hhoommee.. hhaappppeenn iiff tthhee rriiddee ssppuunn eevveenn ffaasstteerr?? PPeeooppllee aarree hhuunnggrryy ttoonniigghhtt.. AAtt tthhiiss rraattee,, II’’llll rruunn oouutt ooff hhoott ddooggss iinn hhaallff aann hhoouurr.. SS cc Miiateeshcisnn tieccosn tttieeehesasetstrsts oe me t t–nbh etihiuextdie aloalenh rcd if vte.pleoeb rupsrn S it s f oadttangoimh rde eepsg r,tr mma oh acmeutkiaoneccr dia hleir isnu d esese,s, ! n a CC j Yo aa MMAA WWAAOOTTRRLLHHDD OOSSFF caacllloc bsuetalsa wc,t iaaooknurnedks ae ttt nodiam d baou oi eunc unstag tteurs ie.es mvQdlline uumccrgasy anuutth tt hlliitnsieaag ts,ott ,f rmiiomaook enn m ati o n. It is impossible to imagine our world without maths. We use it, often without realizing, for PPaanneell ppuuzzzzllee These shapes form a square panel, used a whole range of activities – when we tell the AA in one of the fairground stalls. However, time, go shopping, catch a ball, or play a game. an extra shape has somehow got mixed up with them. Can you work out which This book is all about how to get your maths piece does not belong? brain buzzing, with lots of things to do, many of BB the big ideas explained, and stories about how EE great maths brains have changed our world. FF CC DD 6 000066__000077__118822443388__AA__wwoorrlldd__ooff__mmaatthhss..iinndddd 66 2277//0088//2211 66::0099 PPMM GGuullpp!! TThhee sslliiddee llooookkss TThheerree’’ss aa hheeiigghhtt rreessttrriiccttiioonn eevveenn sstteeeeppeerr ffrroomm tthhee ttoopp.. oonn tthhiiss rriiddee,, ssoonnnnyy.. TTrryy II´´llll bbee iinn tthhiiss qquueeuuee ffoorr II wwoonnddeerr wwhhaatt ssppeeeedd II’’llll ccoommiinngg bbaacckk nneexxtt yyeeaarr.. 1100 mmiinnuutteess,, ssoo II sshhoouulldd ssttiillll bbee ggooiinngg wwhheenn II ggeett ttoo bbee iinn ttiimmee ttoo ccaattcchh tthhee nneexxtt tthhee bboottttoomm?? II wwoonnddeerr wwhhaatt wwoouulldd LLooookk aatt mmee!! II’’mm bbuuss hhoommee.. hhaappppeenn iiff tthhee rriiddee ssppuunn flflooaattiinngg iinn tthhee aaiirr aanndd eevveenn ffaasstteerr?? II’’vvee ggoott ttwwoo ttoonngguueess!! PPeeooppllee aarree hhuunnggrryy ttoonniigghhtt.. AAtt tthhiiss rraattee,, II’’llll OOnnee iinn ffoouurr ppeeooppllee aarree II tthhiinnkk II’’vvee ggoott tthhee rruunn oouutt ooff hhoott ddooggss iinn hhiittttiinngg aa ccooccoonnuutt.. GGrrrr!! II’’mm aannggllee jjuusstt rriigghhtt...... oonnee hhaallff aann hhoouurr.. mmaakkiinngg aa lloossss.. mmoorree ggoo aanndd II’’llll wwiinn aa pprriizzee.. MMAASS cc Mii ateeshcisnn tieccosn tttWWiAAeeehesasetstrsts oe me ta t–nbh netihiuextdie aloaOOlenh rcd if vte.pleoeb rupsrn S it s f oadttTTangoimh RRrde eepsg r,tr mma oh acmeutkiaoneccr dia LLhleir isnu d esese,s, ! HHDD OOSSFF ccaaaacclllllloocc bbssuueettaallssaa wwcc,,tt iiaaaaoooojjkknnuurrnneeddkkss aaee tttttt nnooYYddiiaammoo dd bbaaoouu ooii eeuunncc uuCCCCnnssttaagg tttteeuurrssaaaa iiee..eemmss mmvvQQddlllliiaannee uuuummttccccrrggaaiissyyoo aannuuuuttttnnhh tttt hh..lllliiiittnnssiieeaaaagg ttss,,ootttt ,,ff rrmmiiiioommaaooookk eennnn nuOreesftpewesed nuwa t cttioahnh o y vheasroesM s hloelv pahof e p nwoPPtu ahlywo saat isant o hanettrkknarueiinpndnttmase ggsieeb ns . cfoe sorrabfrpr nesmnn i phr b eassaae ttv htees.r ns, SS hhU aan sdppperaseescetussetna s.ahsnhi ienedYsn i ldoco anplfu dr sgut eendhsuisaei eshn egoatgfdw n po t tomeraroims lna ctkdyaak thnkynah seoir d g tonwa ogu amnc–br de oes.ua tt e It is impossible to imagine our world without maths. We use it, often without realizing, for PPaanneell ppuuzzzzllee PPrrooffiitt mmaarrggiinn AA ggaammee ooff cchhaannccee These shapes form a square panel, used It costs £144 a day to run the dodgems, Everyone loves to try to knock down a whole range of activities – when we tell the AA in one of the fairground stalls. However, accounting for wages, electricity, a coconut – but what are your chances time, go shopping, catch a ball, or play a game. an extra shape has somehow got mixed transport, and so on. There are of success? The stall owner needs to up with them. Can you work out which 12 dodgem cars, and, on average, know so he can make sure he’s got This book is all about how to get your maths piece does not belong? 60 per cent of them are occupied each enough coconuts, and to work out how brain buzzing, with lots of things to do, many of BB session. The ride is open for eight much to charge. He’s discovered that, on hours a day, with four sessions an average, he has 90 customers a day, each the big ideas explained, and stories about how EE hour, and the fare is £2 per session. throwing three balls, and the total great maths brains have changed our world. FF How much profit is the number of coconuts won is 30. So what is CC owner making? the likelihood of you winning a coconut? DD 6 7 000066__000077__118822443388__AA__wwoorrlldd__ooff__mmaatthhss..iinndddd 66 2277//0088//2211 66::0099 PPMM 000066__000077__118822443388__AA__wwoorrlldd__ooff__mmaatthhss..iinndddd 77 2277//0088//2211 66::0099 PPMM MMaatthhss bbrraaiinn 000088__000099__118822443388__MMaatthh__BBrraaiinn..iinndddd 88 2277//0088//2211 66::0099 PPMM MMaatthhss bbbbrrrraaaaiiiinnnn 000088__000099__118822443388__MMaatthh__BBrraaiinn..iinndddd 88 2277//0088//2211 66::0099 PPMM 000088__000099__118822443388__MMaatthh__BBrraaiinn..iinndddd 99 2277//0088//2211 66::0099 PPMM Meninges Cerebrum Where thinking occurs LLooookkiinngg iinnssiiddee Protective layers and memories are stored that cushion the This cross-section of the skull brain against shock Corpus callosum Links the reveals the thinking part of the two sides of the brain brain, or cerebrum. Beneath its Skull Forms a tough casing Hypothalamus Controls outer layers is the “white matter” around the brain sleep, hunger, and body temperature which transfers signals between different parts of the brain. Cerebellum Helps Pituitary gland Controls the release of hormones to control balance and movement AA BBRRAAIINN OOFF TTWWOO HHAALLVVEESS The cerebrum has two hemispheres. Each deals mainly with the opposite side of the body – data Thalamus Receives from the right eye, for example, is handled in sensory nerve signals and sends them on the brain’s left side. For some functions, Medulla Controls to the cerebrum including maths, both halves work breathing, heartbeat, together. For others, one half is blood pressure, and vomiting more active than the other. LLEEFFTT--BBRRAAIINN SSKKIILLLLSS LLaanngguuaaggee The left side of your cerebrum is The left side handles the meanings responsible for the logical, rational of words, but it is the right half that aspects of your thinking, as well as for puts them together into sentences grammar and vocabulary. It’s here that and stories. you work out the answers to calculations. SScciieennttiiffiicc RRaattiioonnaall tthhoouugghhtt tthhiinnkkiinngg Thinking and reacting in a Logical thinking is the job rational way appears to be of the brain’s left side, but mainly a left-brain activity. most science also involves It allows you to analyze a the creative right side. problem and find an answer. MMaatthheemmaattiiccaall WWrriittiinngg sskkiillllss sskkiillllss The left brain oversees Like spoken language, writing numbers and calculations, involves both hemispheres. The while the right processes right organizes ideas, while the left shapes and patterns. finds the words to express them. Left visual cortex Processes signals from the right eye MMEEEETT YYOOUURR Your brain is the most complex organ in your body – a spongy, pink mass made up of billions of microscopic nerve cells. Its BBRRAAIINNlargest part is the cauliflower-like cerebrum, made up of two hemispheres, or halves, linked by a network of nerves. The cerebrum is the part of the brain where mathematics is understood and calculations are made. 10 001100__001111__118822443388__MMeeeett__yyoouurr__bbrraaiinn..iinndddd 1100 2277//0088//2211 66::0099 PPMM