Challenge ‘20 – The Solutions

University of Southampton

Entries

884 entries

32 participating schools

66 prizes awarded

Top Schools

School Number of prizes

Wilson’s School 18

The Westgate School 10

Priestlands School 4

Brookfield Community School 3

Well done to all schools!

1. Barry’s Blocks

Barry has 4 wooden identically sized and shaped blocks. 2 are blue, 1 is red and 1 is green.

How many distinct ways can Barry arrange the 4 blocks in a row?

Barry’s friend Billie is colour-blind, and cannot distinguish between red and green. How many of Barry’s distinct arrangements would Billie see as different?

1. Barry’s Blocks

There are 12 ways of arranging the 4 blocks.

Since Billie cannot distinguish between red and green, he sees this.

The two arrangements on each row are now indistinguishable, so he can see 6 different arrangements.

2. Peter’s Patients

Peter the optician sees 16 patients one Tuesday. 4 are neither short nor long sighted. The total number of short sighted patients is two more than the total number of long sighted patients. How many patients can be short sighted, how many can be long sighted and how many could require varifocal lenses?

2. Peter’s Patients

Short sighted

Varifocals

Neither

Long sighted

4

7 0 56 2 45 4 34 6 23 8 1

2 10 0

3. Light Work

Oscar has 6 tubular neon lights in order to spell out the word VISION to promote his lighting business. It costs £2 to add a corner to a tube, £3 to wire the ends of two tubes together (or the two ends of one tube). It is free to add smooth curves or to stretch the tubes. How much does it cost Oscar to spell the word VISION? Which capital letters of the alphabet is it possible to make with a single tube using the Arial font? Justify your answer.

3. Light Work

Letter V I S I O N Total

Cost £2 0 0 0 £3 £4 £9

3. Light WorkLetter A B C D E F G

Possible?

Letter H I J K L M NPossible?

Letter O P Q R S T UPossible?

Letter V W X Y ZPossible?

4. Window CleaningA skyscraper has 100 floors, numbered 0 (ground floor) to 99, each with one window. A team of 10 inefficient window cleaners are hired to clean the windows. They work from bottom to top and, when each cleaner has cleaned their highest floor, they leave. The 1st cleaner cleans every 10th window (so her first window is on the 9th floor), the 2nd cleans every 9th window, the 3rd cleans every 8th window and so on until the 9th cleans every 2nd

window. The final cleaner has an attack of vertigo and can’t work. Which floors will still have dirty windows?

4. Window Cleaning0 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 27 28 29

30 31 32 33 34 35 36 37 38 39

40 41 42 43 44 45 46 47 48 49

50 51 52 53 54 55 56 57 58 59

60 61 62 63 64 65 66 67 68 69

70 71 72 73 74 75 76 77 78 79

80 81 82 83 84 85 86 87 88 89

90 91 92 93 94 95 96 97 98 99

These numbers are 0, and then 1 less than each prime number that is

greater than ten

5. Spot the DifferenceTwins can be identical boys, identical girls,

non-identical boys, non-identical girls or

one boy and one girl.

What are the possible combinations with triplets?

What are the possible combinations with quadruplets?

5. Spot the Difference

Identical girl

Identical boy

Non-identical girl

Non-identical boy

5. Spot the Difference

5. Spot the Difference

6. Schedule Shenanigans Nick the TV scheduler is writing the Christmas Eve schedule from 6pm to midnight. These are the shows to be broadcast:• 3 fifteen-minute News bulletins; there must be at least 2½ hours between bulletins• 'Night of the Zombie Reindeer', a 2½-hour film which must be shown after the 9pm

watershed • a 30-minute astronomy programme – ‘Stars on TV’ • a 60-minute Christmas special of ‘The Liver Robins’• the Christmas episode of Eastdale Street (30 mins)• a 45-minute sci-fi programme (Snowmen in Space). His wife, Mary, an avid Eastdale Street fan, has asked him to ensure that it doesn’t start before 8pm so she can be home in time to watch it live. Meanwhile, their young daughter Holly has asked him to put The Liver Robins on early enough for her to watch it before her mum puts her to bed when she gets in.Assuming no advert breaks, how could he fit it all in?

Ryan Pippin, the presenter of ‘Stars on TV’, gets embroiled in an academic scandal and the programme needs to be pulled from the schedule. It is to be replaced with 3 ten-minute programmes called ‘Santa Watch’. These need to be spread as evenly as possible through the night, whilst keeping the news bulletins at least 2½ hours apart. How can the schedule be rearranged?

6. Schedule Shenanigans 1800 1805 1810 1815 1820 1825 1830 1835 1840 1845 1850 1855 1900

1900 1905 1910 1915 1920 1925 1930 1935 1940 1945 1950 1955 2000

2000 2005 2010 2015 2020 2025 2030 2035 2040 2045 2050 2055 2100

2100 2105 2110 2115 2120 2125 2130 2135 2140 2145 2150 2155 2200

2200 2205 2210 2215 2220 2225 2230 2235 2240 2245 2250 2255 2300

NEWS

NEWS

NEWS?

NEWS? Night of the Zombie Reindeer

Night of the Zombie Reindeer

Night of the Zombie Reindeer

NEWS

Eastdale StreetStars on TV

The Liver Robins

Snowmen in Space

Eastdale Street must start at 2000 or later

The Liver Robins must finish by 2000

There are 6 possible arrangements of these 4

programmes, any of which is a valid solution.

This fixes the middle news bulletin at 2100

Night of the Zombie Reindeer is 2 ½ hours long, and must be after

2100.

Again, since nothing is shorter than 15 minutes, it must finish

just before the final news bulletin, so it starts at 2115.

The middle bulletin must be between 2045 and 2115.

Nothing shorter than 15 minutes, so must start at 1800.

Similarly the last one must finish between 2345 and 0000, so will

have to start at 2345.

News – 3 fifteen minute bulletins at least 2 ½ hours apart.

That’s 5 ¾ hours so first one must start between 1800 and 1815.

6. Schedule Shenanigans 1800 1805 1810 1815 1820 1825 1830 1835 1840 1845 1850 1855 1900

1900 1905 1910 1915 1920 1925 1930 1935 1940 1945 1950 1955 2000

2000 2005 2010 2015 2020 2025 2030 2035 2040 2045 2050 2055 2100

2100 2105 2110 2115 2120 2125 2130 2135 2140 2145 2150 2155 2200

2200 2205 2210 2215 2220 2225 2230 2235 2240 2245 2250 2255 2300

NEWS

NEWS

Night of the Zombie Reindeer

Night of the Zombie Reindeer

Night of the Zombie Reindeer

NEWS

Eastdale Street

Santa Watch

The Liver Robins Snowmen in Space

Santa Watch

Santa Watch

NEWSS in S

The Liver Robins

So, there are 4 possible arrangements of Eastdale Street,

the final News and the middle and final Santa Watch

programmes, any of which is a valid solution.

Eastdale Street must now start at 2010 or 2020, with the middle Santa Watch after or before it

respectively.

That leaves Snowmen in Space to be on between 1925 and 2010.

The Liver Robins must start at 1825, as there is no programme that can fit in before it so that it

finishes before 2000

The Night of the Zombie Reindeer must now start at 2105.

The middle News bulletin must then start at 2050.

The first Santa Watch is after the first News bulletin, as the middle

News cannot start later than 2050.

The last Santa Watch is before or after the last News bulletin.

7. Second SightZelda the psychic has minimal psychic powers. When she tries to ‘see’ which card Hilda is holding, she gets the colour correct 100% of the time. She can also tell if it is a picture card (King, Queen, Jack) or not. Being a romantic, when a card is a Heart, her chances of correctly identifying the face value are doubled; however, she hates gardening, so, when a card is a spade, her chances of finding the right face value are halved. What is the probability that she guesses correctly if Hilda is holding the Ten of Diamonds, the Jack of Clubs, the Queen of Hearts or the Ace of Spades?

7. Second Sight

Ten of Diamonds: 1/52

She knows it’s red – 1/26

She knows it’s not a picture card – 1/20

Jack of Clubs: 1/52

She knows it’s black – 1/26

She knows it’s a picture card – 1/6

7. Second Sight

Ace of Spades: 1/52 She knows it’s black – 1/26She knows it’s not a picture card – 1/20She hates gardening so 1/40

Queen of Hearts: 1/52She knows it’s red – 1/26She knows it’s a picture card – 1/6She is a romantic so 1/3

7. Second Sight

Ace of Spades 1/40

Queen of Hearts 1/3

Ten of Diamonds 1/20

Jack of Clubs 1/6

8. Go With the FloFlorence Nightingale, born in 1820, is moving around a hospital, and is trying to avoid the administrator who is a terrible bore. It is a dark and stormy night, and she won’t go outside.

On the map on the next slide, each square represents a room. Thin lines are walls with a door in them, thick lines are walls without a door in them.

Each time Florence moves into an adjacent room, the administrator also moves into an adjacent room; however, he is superstitious and so he does not re-enter a room he has just left, although he can return to a room later.

She starts on the ward (W), whilst he starts in the admin office (A).

How might she be able to reach her quarters (Q) without being caught by him?

8. Go With the Flo

N

W E

S

A Q

W

8. Go With the FloN

W E

S

A Q

W

The yellow squares are ones the administrator reaches after an odd number of moves.The white squares are ones Florence reaches after an odd number of moves.Florence will be in a yellow square, and the administrator in a white square after an even number of moves.

This means they will never be in the same room at the same time.The only way for them to meet is in a doorway.

Thank you for watching!