Challenge ' 17 – The Solutions

1. Challenge '17 – The Solutions Sponsored by

2. Entries • 2,252 entries • 95 schools • Number of entries from each school ranged from 1 to 164 from Preston Muslim Girls School • 132 prizes and certificates

3. Top Schools

4. 1. A Chain of Events Three friends, Davina, Jolene and Katie, set out to the Thorpe Towers theme park, travelling on their three bikes. They lock up the bikes and go to enjoy their day. When they return, they find two of the bikes have disappeared. Assuming only one person can ride the remaining bike at a time, and that it is 12 miles back home, how should they proceed, to be fair to each other? The bike has a speedometer with a mileage counter. What if only one bike had been stolen?

5. 1. A Chain of Events • With only one bike left: • Jolene rides for 4 miles whilst the other two walk • Then Katie rides for 4 miles whilst the other two walk • Finally, Davina rides for 4 miles whilst the other two walk.

6. 1. A Chain of Events • With TWO bikes left: • Davina walks for 4 miles whilst the other two cycle • Then Jolene walks for 4 miles whilst the other two cycle • Finally, Katie walks for 4 miles whilst the other two cycle.

7. 1. A Chain of Events “They should sell the remaining bike and use the money to pay for a taxi” “The one whose bike is left should cycle home and send their parents to pick up the other two” “The third girl should ride on the handlebars / pegs”

8. 1. A Chain of Events “Fairest”

9. 2. Lap It Up Alyssa challenges Martin to a race around the go-kart track at the fairground. A race consists of 7 laps. It turns out that Alyssa is a more gung-ho driver than Martin and she completes each lap in 45 seconds, compared with Martin’s minute and a half. If they start off alongside each other, how many times are they alongside each other in total during the race?

10. 2. Lap It Up

11. 3. Connect Twenty-Four Ken has two standard white Lego bricks, as shown. There are 24 unique ways in which Ken can connect his two Lego bricks. Show these 24 arrangements of his bricks. (Arrangements are regarded as the same if one can be rotated to give the other.)

12. 3. Connect Twenty-Four

13. 3. Connect Twenty-Four

14. 3. Connect Twenty-Four

15. 4. St Sebastian’s Shuttlecocks Traditionally, the annual end-of-season badminton tournament at St Sebastian’s High School features 32 pupils. This year, the tournament has attracted 54 entrants. Members of the school team are always automatically admitted to the tournament. It is decided to play one qualifying round during the week before the tournament for all of the other entrants. How many matches take place in the qualifying round? How many members does the school team have? How many matches in total (qualifying plus tournament) will have taken place by the time a champion is crowned?

16. 4. St Sebastian’s Shuttlecocks To get from 54 to 32 players, we need to eliminate 22 players This means 22 matches

17. 4. St Sebastian’s Shuttlecocks 22 matches involve 44 players This leaves 10 players with a bye into the first round, so there are 10 players in the school team.

18. 4. St Sebastian’s Shuttlecocks There are 16+8+4+2+1+22=53 matches altogether. To get one winner, you need 53 losers.

19. 5. Shake it Up The headteacher insists that, at the start of the St Sebastian’s High School badminton tournament, each player must shake hands with every other player. 32 pupils are competing, so how many handshakes will there be?

20. 5. Shake it Up Each player shakes hands with 31 other players 32×31=992 But each handshake involves 2 players, so 992 ÷ 2 = 496 handshakes

21. 6. Basket Balls In a sideshow game at the fair, in order to win a cuddly prize, players have to accurately throw 30 balls into five empty baskets, laid out within a 3 × 3 grid, so that the number of balls in each row, column and each of the two diagonals is the same. Joel decides to try. The four corner baskets contain balls already, as shown opposite: Help Joel win by working out how many balls he should throw into each of the five empty baskets.

22. 6. Basket Balls 30 balls to add plus the 24 already in the baskets gives a total of 54. 54 ÷ 3 = 18 total for each row, column or diagonal. So 6 in the centre. Then we can fill in the other four numbers.

23. 7. Game of Throws In the game of darts, you can hit each of the numbers 1 – 20. Your score is doubled by landing in the outer ring and trebled by landing in the inner ring. You score 25 by hitting the outer bull’s eye and 50 by hitting the inner bull’s eye Jo is practising ahead of a big match. She is a good player, so all of her throws successfully land on the board! List all of the scores, between 1 and 180, which Jo cannot achieve with three darts, showing how you worked this out. (Disregard the rule about finishing with a double.)

24. 7. Game of Throws With one dart:

25. 7. Game of Throws With two darts:

26. 7. Game of Throws With three darts:

27. 7. Game of Throws So that leaves us with: 1, 2, 163, 166, 169, 172, 173, 175, 176, 178, 179 As the scores which cannot be made

28. 8. Four Go to the Fair Four university friends – Chris, Lonnie, Philippa and Robert – go for a summer’s day out at the fair. In some order, they originally hail from Chester, Liscard, Prescot and Rhyl. They go on four different attractions: the carousel, log flume, pirate ship and rollercoaster. However, for each attraction, one student doesn’t ride it and instead takes a turn waiting with their bags. No initial letters of name, hometown or missed ride match. Robert doesn’t want to wet his trendy hairstyle. Chris is too tall for the rollercoaster. Lonnie isn’t Welsh. Match each student with their missed ride and where they come from.

29. 8. Four Go to the Fair Robert doesn’t want to wet his trendy hairstyle. Chris is too tall for the rollercoaster. Lonnie isn’t Welsh. No initial letters of name, hometown or missed ride match.