the big brain workout - telegraph · 2020. 9. 30. · simple sugars increase the serotonin in the...

THE BIG BRAIN WORKOUT

Improve your memory and strengthen your mind

PART 1

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3The Big Brain Workout

IntroductionWelcome to part one of The Big Brain Workout, The Telegraph’s guide to keeping your brain healthy. In this booklet, you’ll find more than 50 puzzles to test your IQ, numeracy, visual awareness and logic. All are designed to enhance your mental agility and improve your memory.

There are useful tips throughout the booklet to help sharpen your skills and prepare you for psychometric tests used by some employers, plus a cryptic Telegraph crossword to challenge your deductive language skills. There’s even a diet and lifestyle section that gives advice on how nutrition and hydration affect your cognitive capacity.

If you get stuck, there are tip boxes throughout and you can always look up the answers at the back of the booklet.

Don’t miss part two of The Telegraph’s Big Brain Workout in tomorrow’s newspaper. It contains more creative, verbal and memory puzzles to test your reasoning (and patience), plus seven Sudoku grids to help improve your numerical, visual and critical-thinking skills.

Good luck!

P.S. Visit telegraph.co.uk/brainworkout for additional lifestyle tips, offers and more.

ContentsIntroduction 3

Diet, exercise and lifestyle 4

The workoutsIQ workout 6Numerical thinking 12Logical thinking 16Visual thinking 20Cryptic crossword 24

AnswersIQ workout 26Numerical thinking 28Crossword 28Logical thinking 29Visual thinking 30

DON’T MISS THE SECOND PART OF THE BIG BRAIN WORKOUTThe second brain-training booklet in tomorrow’s newspaper contains more than 50 IQ, creative and numerical puzzles, as well as a Toughie Telegraph crossword. There are also more diet and lifestyle tips to help keep your brain functioning at capacity.

Over

50 questions

4 The Big Brain Workout

DIET, ExERCISE & lIFESTylEYour brain responds to what you eat and drink, and it can adapt its structure depending on how often you exercise, drink alcohol or smoke. B vitamins, for example, help form new neurotransmitters. They can be found in berries, spinach, green cabbage, broccoli and watercress.

Fish, eggs and other sources of protein make a good breakfast, although you should avoid breads and cereals unless they are wholegrain. Drink fruit juice for the vitamins and water to stay hydrated.

For lunch, try oily fish like mackerel and salmon with salad or raw vegetables. If you must have dessert, fresh fruit is preferable to biscuits and pastries.

Eat pasta and wholemeal breads or cereal in the evening to provide energy for the following day. Simple sugars increase the serotonin in the brain and may help you sleep.

Avoid saturated transfats as they are difficult for the body to break down, deprive the brain of oxygen and inhibit the uptake of essential unsaturated fatty acids.

Boost cognitive function and improve your memory with folic acid and selenium in dark leafy vegetables, seafood and wholegrain breads.

Boron, iron and zinc can also improve mental capacity. Boron can be found in pears, peaches and grapes; zinc in fish, meats, cereal and whole grains; and the iron in meat and fish is more readily absorbed by the vitamin C in citrus fruits and salad.


The amount of sleep people need varies, but it’s usually from five to eight hours. Quality rather than quantity is more important for brain health, although this can be affected by babies, stress or snoring. Try to remove televisions and radios from the bedroom (or turn them off completely) and use breathing exercises to relax.

Alcohol reduces the body’s stores of vitamin B1, which affects our ability to think. It also leads to overconfidence, distraction and error-proneness, results in dehydration and contributes to short-term memory loss.

Smokers seem convinced that a cigarette keeps them mentally alert but smoking reduces the oxygen-carrying capacity of the blood, limiting the amount of oxygen that reaches the brain. In the long-term, lung damage severely restricts the amount of oxygen the blood can absorb.

Anaerobic exercise produces serotonin, which relaxes you and helps you sleep. Exercises that involve controlling your breathing – like holding yoga positions – typically produce elongated brainwaves with higher peaks.


IQ WORKOUT

Q1. Boxes and coinsEach label is incorrect. You need to re-label each box correctly after removing a single coin from one box without looking inside.

Q2. The spider and the flyWhat is the shortest distance the spider can walk to reach the fly (the room is 30m × 12m × 12m)? The spider is 1m from the floor. The fly is 1m from the ceiling.

12m

30m

12mAll copper coins

Copper andsilver coins

All silver coins

Introduction There are several kinds of intelligence, most of which continue to develop during our lifetimes. Unfortunately, many employers still use traditional verbal, numerical and spatial IQ tests to assess an applicant’s suitablity for a role, which, while they may be important when discussing overall intelligence, ignore creativity, memory, the ability to predict behaviour and general knowledge. These skills improve as you age, and life satisfaction has been found to be more closely linked with emotional intelligence than high IQ scores. The following puzzles will test your mental agility and reasoning.


Q3. Getting to the root of the problemA garden has 12 trees in a row. How could they be uprooted and replanted to make six rows with four trees in each row?

Q4. Give me one honest personThe man in the road needs to find one honest person. He knows that one side of the road is Truth Land where everyone always tells the truth, and the other side is False Land where everyone always lies. He asks Mr A to ask Mrs B which side of the road she lives on. Mr A returns saying that Mrs B says she is from Truth Land. Is Mr A honest?

Truth Land

A

B

False Land


Q5. CashYou are a cashier and one customer has shaved 1g of gold off each of the gold coins in one of their 10 bags. All the bags contain the same number of coins. How can you find the bag with the shaved coins in one weighing?

Q6. Eight small rectanglesRemove four lines to leave eight small rectangles.

Is it wise to eat sage?Apparently, if you rip sage leaves into small pieces and add a teaspoon of the ripped leaves to a cup of boiling water, you will feel the benefit if you sip the drink during examinations, tests or presentations. Sage inhibits the breakdown of acetylcholine, which the brain uses when it’s working.


Q7. What time is it, Mr Wolf?Which clock, A, B, C or D, completes the series?

Q8. Stacking pyramidsMove three lines only to make five triangles.

Series

IQ WORKOUT

A B C D


Q10. Topsy-turvy wordsWhen you insert the missing letters you will discover a way of thinking that makes you more intelligent.

Q11. The one-match trickMove one match to balance the equation.

NT

IE

LY

IE

NK

YE

ON

WO

AT

RE

AP

IP

CE

OT

AY

Q9. Where next?Where should the next • go (given that the first • was placed in the top triangle)?


Q12. Half full and half emptyHow can you change this line of seven glasses so that they alternate full and empty? You may only touch one glass.

Q13. Water bottlesThree Bedouins travelling together in the desert decided to go their separate ways. Between them they had seven full, seven half full and seven empty water bottles. The bottles each held five litres. They decided to divide their water and the 21 bottles equally between them. How did they do it?

Q14. What day is it?In summer, water lilies double their area every 24 hours, so that on day 30 the pond is covered. The first lily appeared on the first day of the month and the pond is now half covered. What day of the month is it?

Can you boost your IQ?Learning different skills helps build new neuron connections, and practising old skills leads to thicker myelin insulation on each pathway. What you learn will increase your general knowledge and capacity for creative thinking.

Q15. Refusing to pay for bad serviceA restaurant’s royal customer waited 43 minutes for the waiter to take her order. Finally, she got up and left, so the manager asked if she had paid her bill. She shook her head and wrote this explanation:

10004180204

What did she mean?

IQ WORKOUT


NUMERICAl

Q1. The Tutonton pyramidThis is a sketch of the profile through the centre of a pyramid. It has a square base with an area of 81 square metres. Each block weighs approximately 1,212 kg.

How tall is the pyramid? How broad is the pyramid? How many metre-cube blocks were needed to build the Tutonton pyramid? And how much does it weigh?

Q2. Balancing actHow many triangles will balance scale C (distance from fulcrum is not a factor)?

Introduction If you work in business, it seems self-explanatory that you will need to be numerate as you will be analysing profits, losses, balances, returns and cash. It may be less obvious why those working in the public service or charity sectors need numeracy skills, but basic maths is essential for quality control, project management and service planning.

While the ability to make meaningful estimates lies at the heart of maths, accurate mental arithmetic is more important as engineers, accountants, architects and scientists all work from precise figures, while fund managers, bankers and analysts need to make forecasts relating to the economy.

Practising your times tables, plus basic addition, subtraction and division, can help revitalise dormant neural connections within the brain, increasing intelligence, memory and cognitive capacity. The questions on these pages will test numeracy and reasoning.

A B C

THINKING


Q3. A distance too farHow far away is London?

Q4. Cracking the codeFill in the blank circles.

Q5. Clocking onIf A = 3 o’clock, what time will clock F show?

York 50km

Oslo 60km

Paris 70km

Madrid 80km5 1 4

1 3 6 5 8

3

A B

E

C

F

D


Q6. Roll out the barrelYou only have a torch to determine whether a wooden barrel is more than or less than half full. How will you do it?

Q8. Seeing spotsThis is the top die on a single stack of 10 dice. How many dots are visible on the stack?

Q9. The magic shrinking fluteA musician doesn’t want to check his precious 48cm flute in as hold luggage, but an officious security officer is limiting the length of hand luggage to 40cm. How can the musician legally carry his flute on board as hand luggage?

Q7. Sands of timeYou need to measure a quarter of an hour and you have two sand glasses. One takes 11 minutes to empty, the other seven minutes. How can you use them to measure your quarter hour?

SleepMost people have a natural sleep time that varies between five and eight hours. If you sleep for less than this natural time, your performance in IQ tests is likely to fall. If this pattern continues for a few nights, your IQ can fall by up to 15 points. This can impair your ability to learn important information, as well as adversely affecting your ability to drive or react to unusual circumstances. A study at Southampton University found that maintaining a sleeping routine was beneficial for cognitive function, while the University of Perth has backed up the claim that afternoon power naps can improve performance.


NUMERICAL THINKING

Q10. All the threesPut a 2 in the centre of the grid below and then select from only three other numbers, other than 2 and 3, to make each row, column and diagonal add up to 10. Which three numbers will work?

Making number skills countNumeracy can be improved by playing games like backgammon, cards, darts and dominoes, as well as by comparing prices by weight in supermarkets or by splitting the bill in a restaurant. The skills involved in games like backgammon help the brain recognise numerical patterns, and learning the scoring system in darts helps us to add, subtract and multiply large numbers with confidence. Mastering basic maths can help us compute, estimate and quantify. These are the skills that develop the parts of the brain dealing with memory and logic.

3 3

33

33

3

3

3

3


lOGICAl

Q1. Brazil for the cup?Consider the following, which was overheard on the street: “Brazil have a better chance of winning than Germany or France, or even Argentina. Therefore, they will probably win the World Cup.”

A. Is this statement deductively valid?B. Is this statement inductively

reasonable?C. What useful change could you make

to the statement?

Q2. English football fans abroadEnglish fans will probably cause trouble at the World Cup so England could be expelled, maybe not from the World Cup itself but certainly from the next European Championships. Therefore, it is unlikely that England will be playing in the next European Championships.

A. Is this deductively valid?B. Is this inductively reasonable?C. How could you write it as an

inductively reasonable argument?D. How would you need to write it as

a deductively reasonable argument?

Q3. Theatrical confusionWhen you say “go nurse”, which medic stirs?

Q4. Moon starerAt certain times of the month he (or she) may be a moon starer.

Q5. HearingCan you hear four vowels and a D, if not ‘goodbye’?

Introduction The foundation of logic is based on the three I’s, which shows the way we move from Information via Inferences to Implied action. Applied thinking involves using critical, creative and reflective reasoning to assess whether the information we receive is true. We can then process this information, turning it into knowledge on which we can act. We can use critical thinking to assess whether inferences are reasonable, but we must use caution to verify the accuracy and relevance of information before forming an opinion.

In deductive logic, reasonableness = a justified belief + a valid conclusion.

In inductive logic, reasonableness = a justified belief + a probable conclusion.

THINKING


Q6. letter puzzleThese letters ZAyxWBCDEFGVUTSRQHIPJONMK scream Christmas.

Q7. Can you throw any light on this?Which switch controls which light? The switches are outside the room and you can only enter the room once. How will you find out?

Becoming a critical friendYou should try to develop your creative, critical and reflective thinking. Critical thinking is not as destructive as it sounds. The aim is to question the information you receive, and the person delivering it, so that you can use the knowledge to reach more accurate conclusions.


Q1 1. And the last shall be… last Runner 4 is faster than runner 1. Runner 2 is faster than runner 3. Runner 3 is slower than runner 1. Who will be last?

Q1 2. Who’s lying? Larry always lies. Tommy always tells the truth. Which one of them said of the other, “He claims he’s Larry.”?

More likely than notApplied thinking is the kind of thinking that allows you to turn information into knowledge on which you can act. In the practical world, inductive logic is more useful than deductive logic.

Q10. Board meetingHow many queens can you place on a chess board without one threatening another? (Queens can move horizontally and on the diagonal.)

Q13. Have room for a DB5 VolanteA collector of vintage cars wants to expand his collection so he increases the size of his air-conditioned garage by 50 per cent. Each car on average is 15ft (4.5m) long and 6ft (2m) wide. During the auction season, the collector acquires four more cars and fills the garage. How many cars does the collector have now?

Q9. Will you blow it?The bridge will be blown up at midnight. It is now 22.50. You can only cross with a torch. There is only one torch and only two people can share it at any time. You are fit and it only takes you four minutes to cross the bridge. Your colleagues are slow: B takes eight minutes, C 30 minutes and D 40 minutes. How can you get everyone safely across the bridge before midnight?

Q8. Is this fair?If he were to give her £300 of what he earns per month, they would earn the same, but if she were to give him £300 of what she earns, he would earn five times what she earns? What do they each earn?


LOGICAL THINKING

Past, present and futureApplied thinking allows you to make decisions that other people see as intelligent. It helps you to learn new skills, solve problems, assess risk and make practical plans for the future. These plans are based on your reading of situations to form valid opinions. You can improve your ability to think by questioning the accuracy of information. This involves uncovering assumptions, questioning the validity and motivation of sources, and checking your perceptions against those made by others.

Through creative and reflective thinking, you can list insights, interpretations, principles, lessons, ideas and conclusions. You can then test your hypotheses using deductive and inductive logic. If your plans are affordable, ethical and likely to be effective, they can then be implemented.

Q14. lord of the ring cycleWhen Richard Wagner enrolled for a course, he was asked a question, to which he replied “9-W”. What was the question?

Q15. The brain trainThe brain trains run from A to B, one a minute, 24 hours a day. The return journey at the same speed takes one hour. How many trains do you pass on the journey from A to B?

Q16. The Monty Hall problemYou reach the final stage of a gameshow and the host shows you three doors. Behind one is a car, behind the other two, nothing. The host asks you to pick a door to try to win the car. You choose door A. The host then opens door B to reveal nothing. The host now offers you the chance to stick with your original choice (A) or switch to door C. He knows what is behind each door but he’s acting in good faith and is not trying to influence your decision. What should you do?


VISUAl

Q1. Which kite will fly?Which of these box kites will fly?

R X XE

O

O X

E

O E

X

O X

R

X

E

X R

View A View B View C View D View E

Introduction Visual processing is one of the brain’s most important functions. Research shows that even those who are visually impaired can maintain visual brain health by imagining images. Visualisation also plays an important role in improving memory, creative thinking, ethical thinking and empathy.Reflective thinking relies on you perceiving what you saw, heard and felt to be real. It helps you turn those experiences into lessons and ideas that you believe to be true, and which allow you to formulate the actions that guide how you behave in the future. Predictive thinking draws on many skills, the most important of which is visual thinking.

Problems and puzzles involving visualisation and prediction are the bread and butter of serious brain-trainers. Those in this section aim to test how visual thinking helps prediction and reflection.

THINKING


Q2. Pack it inWithout using a calculator, how many boxes of size 3 × 3 × 6 can you pack into these containers?

Q4. Square things upArrange these pieces into a square with no spaces.

Q3. Which way?What is the shortest route from A to B?

33

6

30

30

65

B

3

365

A

300

300600

C

B

A

1

1

1 2 2 3 3 4

Visual and spatial IQEmployment applications usually involve psychometric tests that assess your visual and spatial aptitude. Visual IQ is considered an important predictor of overall intelligence. You can improve your visual aptitude by practising questions like those listed here.


Q5. Gone dottyHow many dots would you see altogether as you walked around this stack of giant dice? What is the total of the concealed faces?

Q6. Drawing on experienceMr B has four socks tucked away at the back of his drawer. He only owns black or grey socks. The chance of him pulling out two black ones is 50:50. What are his chances of pulling out two grey socks?

Q7. Bottoms upWould the flat pack below make the cube? If so, what symbol would be on the bottom?

X

Children’s IQRecent research shows that the numerical and verbal IQs of school children have been falling, most likely due to changes in the curriculum and having more screen-based technology at home. For the same reason, our children’s visual IQ has been rising.


VISUAL THINKING

Q8. learner driversThese L drivers have a mind of their own. They have driven from A to B to C. Where will they go next?

Q10. A shocking solutionTen lively rams need to be separated temporarily. You have one long loop and three short loops of electric fencing with plenty of stakes. How can you stake out four circular (more or less) pens that will keep the 10 rams apart?

Q9. Sealed with a loving kissWhere will the next seal go?

?

Creative visualisationWhen you laugh, scream or cry while watching a film, the images may not be real but your emotions are. You can use visual thinking to create or recall images to help you feel the way you want to feel. These images can help you feel excited or calm, creative or comfortable, or they can alleviate the vague despondency that often overcomes people for no apparent reason. Imagining positive situations vicariously can help boost your visual IQ, while recalling details about particular events can help embed long-term memories.

A B C


CROSSWORD

Across5 At present time, ad’s on way, being

redesigned (8)

8 Living quarters round a royal residence (6)

10 Rank: duke, for example, touching end of scale (6)

11 Contribution made by party leading country (8)

12 Spectators at the match will understand what is happening (4,3,5)

15 Room and board for guard (4)17 Apportion a return of tax (5)18 Diamonds shown by crude sketch (4)19 Teenage stats prepared for brokers

(6,6)

22 Fail to hear one about our state (8)24 At home and abroad, to some extent

(2,1,3)

25 Alcoholic drink in hamper, no doubt (6)26 Someone telling jokes about the end

of Robin Hood? (8)

Down1 Navy’s first missile, limited in scope (6)2 To be compassionate, take a card (4,1,5)3 Trip includes Sweden leg (4)4 Fish from lake kept by fool (8)6 Ring article in newspaper about new

evergreen shrub (8)

7 Comical team breaking up (4-9)9 About to argue and boast (4)13 Abroad, long-suffering person who

might be attending hospital (10)

14 Fury surrounding traditional fashion industry (3,5)

16 Friendly, poor farmer, about 50 (8)20 Apprentice finally failing to pass (6)21 Splendid penalty (4)23 Disentangle a French implement (4)

TELEGRAPH CRYPTICIntroductionIf you’re new to cryptic crosswords, this could be the place to start. Most cryptic clues consist of a definition (the answer) and wordplay (clues that together make up the answer). Look out for a word telling you to manoeuvre the letters of the following or preceding word to make an anagram, or double definitions where two words offer different meanings of the answer. You might also find abbreviations like C for ‘caught’ (on a cricket scorecard), H for ‘hearts’ (bridge), B for ‘bishop’ (chess), P for ‘parking’, and T for ‘time’. These may be inserted or added to another word so keep an eye out for ‘in’, ‘by’ and ‘under’.


1 2 3 4

5 6 7 8 9

10 11

12 13

14

15 16 17 18

19 20

21

22 23 24

25 26

Solution on page 28


ANSWERSIQ WORKOUT

Q1. Take one coin from the ‘copper and silver’ box. If it’s copper, the box is all copper and the last two labels must then be swapped.

Q3. Invert one triangle on another. Plant one tree at each apex plus one at each intersection.

Q4. Yes.

Q5. Take one coin from bag 1, two coins from bag 2, three from bag 3 and so on. Weigh the coins you have removed together. The number of grams short is the number of the bag with the shaved coins.

Q7. B. (Hour hand moving 15 minutes clockwise, minute hand moving 15 minutes anticlockwise.)

Q8.

Q9. Count each triangle from left to right, starting with the first • in the top triangle. Count 2 for the next •, then 3, then 4, and so on.

Q6.

Q2. 40m.

1

8

32

7654

12m

40m24m

32m

30m

12m


Q11. Move one match from the equals sign and place it parallel with the minus sign.

Q15. One owes nothing for one ate nothing to owe for.

Q12. Tip the contents of glass three into glass six.

Q13. Pour two half bottles into two other half bottles to make nine full bottles, three half full and nine empty.

Q14. The 29th day. The lilies will double their area overnight to cover the pond completely on the 30th day.

Q10. Applied thinking.

A N T

P I E

P L Y

L I E

I N K

E Y E

D O N

T W O

H A T

I R E

N A P

K I P

I C E

N O T

G A Y


ANSWERSNUMERICAL THINKING

TElEGRAPH CRyPTIC CROSSWORD SOlUTION

Q1. 5 metres; 9 metres; 165 blocks (81 + 49 + 25 + 9 + 1 = 165). As 165 = 2 × 100 × 1,000kg ÷ 1,212, the mass is 200 tonnes.

Q7. Start both sand glasses. Your time starts when the seven-minute glass is empty. Turn the 11-minute glass again when it is empty: 4 + 11 = 15 minutes.

Q8. 146. (Opposite sides of dice always add up to seven, so four sides always add up to 14. Ten dice = 140 plus the six you can see on top.)

Q9. Wrap it as a rectangular package 30cm × 40cm and lay the flute along the diagonal, which will be 50cm.

Q10. 0, 1 and 4.

Q2. Four triangles.

Q3. Vowel = 20km, consonant = 10km. London = 80km.

Q5. 6.35. The clock advances by 2hrs 5mins, then 4hrs 10mins and so on until 32hrs and 80mins. The hands are not independent so the 80mins moves the hour hand on an extra hour.

Q6. Tilt the barrel towards you so that the liquid just comes to the edge without spilling. Shine the torch into the barrel. If you can see the bottom of the barrel, it is less than half full.

Q4. Add 222 to 514 and place the first number of the answer on top of the last two.

3 6

7

N H S FN O W A D A Y S P A L A C E

L R V I I A RD E G R E E D O N A T I O N

A O A E H WK N O W T H E S C O R E

D E P U A RK E E P A L L O T D R A W

R L R I P GE S T A T E A G E N T S

F A T T L RM I S S O U R I I N A W A Y

N A N N E P DP E R N O D G A N G S T E R

T O T E


LOGICAL THINKING

Q1. A. No.

B. No, Brazil could have a better chance than the others without that chance being greater than the 50:50 necessary to justify ‘probably win’.

C. They are therefore more likely than Germany, France and Argentina to win.

Q3. Surgeon.

Q4. Astronomer.

Q5. Audio, Adieu.

Q6. Noel (no L).

Q7. Switch one on for a minute and then switch it off; switch the second one on and enter to find one light on, one off and one warm.

Q2. A. No. B. No.

C. We need a justifiably believable premise, plus a more than 50:50 chance of the outcome. How about: ‘English fans will probably cause trouble at the World Cup, because they usually do in matches against foreign teams. And English fans have been warned that trouble at the World Cup will lead to expulsion from the European Championships. So England will probably be banned from playing in the next Euros.’ This seems like an inductively reasonable argument.

D. We need a justifiably reasonable premise plus an inescapable conclusion for an argument to be deductively reasonable. So, ‘Some English fans are bound to cause trouble at the next World Cup because English fans cause trouble at every non-friendly away match. If English fans cause trouble at the next World Cup, then England will automatically be banned from the European Championships. Therefore England will not play in the next Euros.’

Q8. £600 and £1,200.

Q9. Take B and send B back with the torch. Call for C and D, then take the torch back yourself to collect B with two minutes to spare.

Q10. 8. (Hint: 4, 6, 5, 2, 7, 1, 3, 8.)

Q11. Runner 3.

Q12. Larry. Q13. 12.

Q14. “Do you spell your name with a V?”

Q15. 59 brain trains.

Q16. Switch. You have a 50:50 choice but not a 50:50 chance. You initially have a 1-in-3 chance of picking the door hiding the car. If you stick, your chance of winning remains the same. Two times out of three your initial choice (A) is wrong and the remaining door (C) hides the car. Try it with three cards.


ANSWERSVISUAL THINKING

Q1. D.

Q5. 47; 16 (The opposite faces on a die add up to seven, so 3 x 14 = 42, then add the five. For the hidden faces 2 × 7 = 14, then add the two opposite the five at the top.)

Q7. Yes. A triangle.

Q8. Small L always rotates clockwise by 90°. Middle L never moves. Big L flips.

Q9. Each time the seal is put on the back of the envelope, it moves an extra number of quadrants clockwise.

Q6. Zero. We have to draw out the possible scenarios that would give Mr B a 50:50 chance of picking one black sock and then another black sock. Visualize the possible combinations of black and grey socks:

A. 4 greyB. 3 grey; 1 blackC. 2 grey; 2 blackD. 1 grey; 3 blackE. 4 black

Q2. A = 10; B = 1,050; C = 1,000,000.

Q4. One of many possible solutions.

Q3. There is no single shortest route because the diagram is symmetrical about the AB axis. (Hint: it is often easier than using geometric or trigonometric calculations to lay a piece of thread along the routes or to mark the routes off on the edge of a piece of paper.)

1 1

2

4

33

2

If Mr B is to have any chance of a black pair, we can eliminate A and B straight away. In case C, his chances would be 1/2 × 1/3 = 1/6. In case E his chances are 100 per cent, so a 50:50 chance of picking two black socks must be case D, in which case there is no chance of a grey pair.

12

3

6

9

8 10

5 7

4Q10. One of many possible solutions.

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