Senior Challenge ‘22�The Solutions

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Entries

• 1163 entries
• 92 schools
• Number of entries from each school ranged from 1 to 100 from The Blue Coat School, Liverpool
• 96 prizes and certificates

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Top Schools

63 schools received at least one prize or certificate

1. A Young Girl Reading

Emily is reading a stapled comic, but 4 of the pages are missing.

Pages 6 and 39 are missing.

Which other pages are missing?

How many pages did the comic have originally?

1. A Young Girl Reading

7

5

2

3

4

6

1

6

39

5

40

1. A Young Girl Reading

2

43

4

41

6

39

1

44

2. Napoleon Crossing the Alps

Napoleon’s route through the Alps involved travelling through the Grand St Bernard Pass, a climb of 6km and a descent of 4km. His army travelled twice as fast downhill as it did uphill and the whole journey took 8 hours.

How long did it take him to reach the top?

2. Napoleon Crossing the Alps

3. Composition with Red, Blue & Yellow

Ann is a clever painter. Inspired by Mondrian’s work, she decides to create a mathematical work of art. She divides a square canvas into 9 equal squares and paints the central one red. She then divides each of the 8 remaining uncoloured squares into 9 equal squares, painting each of the central squares so formed yellow. The remaining uncoloured squares are again each divided into 9, the centres this time being painted blue.

This process is repeated using a different colour

for each set of central squares until just over half

the original area of the canvas has been coloured.

How many different colours have then been used

and how many central squares have been painted?

3. Composition with Red, Blue & Yellow

3. Composition with Red, Blue & Yellow

4. The Potato Eaters

Vincent sells potatoes. He doesn’t sell potatoes by weight or one at a time. He insists on selling them in pairs. His customer Graham wants a potato that weighs exactly 200g.

Vincent tells him, “I only have 3 potatoes left.”

“Here they are: A, B and C.”

“A & B together weigh 300g, A & C together weigh 500g, B & C together weigh 400g.”

“You can have any pair of them.”

Which, if any, of the potatoes weighs 200g?

Which pair of potatoes should Graham buy?

4. The Potato Eaters

5. When Did You Last See Your Father?

William replied: “On the first Monday of a certain month last year (2021), I saw him in London, and then, on the Monday after the first Sunday of the month, I saw him in Cardiff.

The following month, we met in Edinburgh on the first Monday of the month and then again in Liverpool on the Monday after the first Sunday of the month.”

Work out the actual dates of William and his father’s meetings in these cities.

Why could this arrangement of Mondays not

have happened the year before last (2020)?

5. When Did You Last See Your Father?

For the Monday after the first Sunday to be distinct from the first Monday, the first Monday must be the 1^st of the month.

For this to happen in two consecutive months, the first month must have 28 days.

This means the first month must be February, and so the four dates are:

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5. When Did You Last See Your Father?

• This would not work in 2020, as it was a leap year and so February had 29 days.

6. Mona Lisa

Lisa moans and complains a lot about her drink being flat.

In fact, her local restaurant manager, Ken, considered banning her for making spurious complaints, but then wondered if he should put her forward for a quality-control job with the fizzy-drinks company.

As evidence, he put together a table showing the last 100 drinks she has bought at the restaurant.

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If Lisa says a drink is flat, how likely is she to be correct?

How good is Lisa at detecting whether or not a drink is flat?

Would Lisa be good at the quality-control job? Justify your answer.

6. Mona Lisa

If Lisa says a drink is flat, how likely is she to be correct?

When Lisa complains, she is correct 5 times out of 8, which is 62.5%

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6. Mona Lisa

How good is Lisa at detecting whether or not a drink is flat?

Lisa correctly identifies 5 out of 6 flat drinks and 91 out of 94 fizzy drinks, this means she was correct 96% of the time.

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6. Mona Lisa

Would Lisa be good at the quality-control job? Justify your answer.

Lisa correctly identifies 96% of the drinks, but the crucial thing here is that she only missed one flat drink.

So, yes she would be good at the job.

7. Arrival of the Normandy Train

Claude is waiting to meet the train from Normandy at Gare Saint-Lazare in Paris. He tells Marie the hour of the train’s arrival and he tells Jean at which minute it arrives. He also tells them both that the train arrives between 0600 and 1000.

They consult the timetable and find the following services between those times:

0620, 0639, 0650; 0717, 0746; 0825, 0839; 0917, 0925, 0950.

Marie then says, “I don’t know when Claude’s train arrives, but I’m sure that neither does Jean.”

Jean replies, “I didn’t know his train, but now I do.”

Marie responds, “Now I do as well!”

When is Claude’s train & how do you know?

7. Arrival of the Normandy Train

Marie then says, “I don’t know when Claude’s train arrives, but I’m sure that neither does Jean.”

06xx and 07xx have unique minutes, so all of these times can be eliminated

Jean replies, “I didn’t know his train, but now I do.”

This eliminates the times xx25.

Marie responds, “Now I do as well!”

This eliminates 09xx, leaving 0839.

8. The Last Supper

On the last night of Scout camp, Chris, his wife Jo and the 11 other adults sit down for supper along one side of a long rectangular table.

Chris takes the central seat, with six adults on each side of him.

Jo must sit next to Chris.

Alex, Barb and Dave insist on being next to one another (in any order) on the side to Chris’s left.

Eddie, Fred and Gareth are fed up with Alex’s snoring, so they refuse to be on the same side of Chris as him.

Heather, Iris, Kane, Luke and Michael have no seating preferences.

How many arrangements of the adults are possible?

8. The Last Supper

Case 1: Jo sits on Chris’s left side.

Consider A, B and D as one ‘block’, because they must sit adjacent to one another. Their block can be seated in 3 positions and their order within the block doesn’t matter, so there are 3! = 6 ways of arranging them for each seating position, giving 18 possible arrangements.

E, F and G must sit to Chris’s right side. Six seats are vacant, so they can be arranged in 6 × 5 × 4 = 120 ways.

Finally, the remaining 5 adults don’t care where they sit, so can be arranged in 5! = 120 ways.

Thus, with Jo on Chris’s left, there are 18 × 120 × 120 = 259, 200 possibilities.

8. The Last Supper

Case 2: Jo sits on Chris’s right side.

Again, consider A, B and D as one ‘block’. Their block can be seated in 4 positions and their order within the block doesn’t matter, so there are 3! = 6 ways of arranging them for each seating position, giving 24 possible arrangements.

E, F and G must sit to Chris’s right side. Five seats are vacant, so they can be arranged in 5 × 4 × 3 = 60 ways.

Finally, the remaining 5 adults don’t care where they sit, so can be arranged in 5! = 120 ways.

Thus, with Jo on Chris’s right, there are 24 × 60 × 120 = 172,800 possibilities.

8. The Last Supper

Adding these together, we find 432,000 ways to arrange the adults.

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Thank you for watching!