Parallelogram 9 • Level 3 • 2 Nov 2023The Secret Maths Club

Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

a portmanteau word combining parallel and telegram. A message sent each week by the Parallel Project to bright young mathematicians.

Tackle each Parallelogram in one go. Don’t get distracted.
Finish by midnight on Sunday if your whole class is doing parallelograms.
Your score & answer sheet will appear immediately after you hit SUBMIT.
Don’t worry if you score less than 50%, because it means you will learn something new when you check the solutions.

1. The secret maths club

3 marks

1.1. There is a secret mathematical club. Before anyone enters, the doorman gives them a number and the mathematician has to respond with the correct matching number.

For example, the doorman says “1” to the first mathematician, who correctly replies “3”.

The doorman says “2” to another mathematician, who correctly replies “3”.

The doorman says “5” to another mathematician, who correctly replies “4”.

The doorman says “7” to you. What should you reply?

Correct Solution: 5

Show Hint (–1 mark)

–1 mark

Instead of thinking about numbers, think about words. So, doorman says “seven”.

You should reply with the number of letters in the word spoken by the doorman. So if the doorman had said "six" or "ten", the reply would have been 3. Had he said "eleven", then the reply would have been 6.

A palindrome is a word that is the same read forwards or backwards, such as “level” or “mum”. One evening, the secret maths club changes its rules, so that the doorman asks questions and the answers have to be palindromes. How would you answer these questions?

1 mark

1.2. Give me another word for midday.

Correct Solution: noon

2 marks

1.3. What does Lewis Hamilton drive?

Correct Solution: racecar

Show Hint (–1 mark)

–1 mark

The answer is a 7-letter word starting RAC_ _ _ _ .

2 marks

1.4. Name something a bit like a canoe?

Correct Solution: kayak

Show Hint (–1 mark)

–1 mark

The answer starts (and therefore ends) with a K.

2. Square wheels

Everyone assumes that you need round wheels for a smooth ride, but that assumes that the road is flat. If the road has very particular surface, then a square wheel will give you a much smoother ride than a round wheel, as this video reveals.

(If you have problems watching the video, right click to open it in a new window)

This clip explains some of the maths needed to make a square wheel work on a bumpy road.

(If you have problems watching the video, right click to open it in a new window)

2 marks

2.1 When describing a smooth ride with square wheels, which of these statements is not true?

Each bump must be shaped like a catenary.
A catenary is the curve formed by a hanging chain.
A catenary is the path formed by a leaping cat.
Another name for a catenary is a hyperbolic cosine.
The length of each arc must be as long as one side of the square.
(Not answered)

3. Junior Maths Challenge Problem (UKMT)

3 marks

3.1 In our school netball league a team gains a certain whole number of points if it wins a game, a lower whole number of points if it draws a game and no points if it loses a game.

After 10 games my team has won 7 games, drawn 3 and gained 44 points. My sister's team has won 5 games, drawn 2 and lost 3.

How many points has her team gained?

28
29
30
31
32
(Not answered)

Show Hint (–1 mark)

–1 mark

Suppose that there are w points for a win, and d points for a draw. Since my team gains 44 points from 7 wins and 3 draws, 7w+3d=44.

Since 7w<44, we have 1≤w≤6.

Also, since 3d=44−7w, then 44−7w must be a multiple of 3.

Suppose that there are w points for a win, and d points for a draw. Since my team gains 44 points from 7 wins and 3 draws, 7w+3d=44. Here w and d are positive integers with w>d.

Since 7w<44, we have 1≤w≤6. Since 3d=44−7w, 44−7w must be a multiple of 3. The only whole numbers in the range 1≤w≤6, for which 44−7w is a multiple of 3, are w=2 and w=5.

When w=2, 3d=44−7w=30, giving d=10, contradicting w>d. When w=5, 3d=44−7w=9, giving d=3. In this case w>d. So w=5 and d=3.

Therefore, my sister’s team with 5 wins and 2 draws has gained 5×5+2×3= 31 points.

4. The juggler

This image was tweeted recently by Matthew Syed, author of books such as “You Are Awesome”, which emphasise how we can all get better at everything if are willing to be positive, dedicate some time and put in some practice. This includes getting better at mathematics.

From the audience's point of view, all they see is a magnificent juggler, but he knows that his performance is the result of a gradual rise, step by step, during which time he had a huge number of failures and smashed hundreds of plates.

Before you hit the SUBMIT button, here are some quick reminders:

You will receive your score immediately, and collect your reward points.
You might earn a new badge... if not, then maybe next week.
Make sure you go through the solution sheet – it is massively important.
A score of less than 50% is ok – it means you can learn lots from your mistakes.
The next Parallelogram is next week, at 3pm on Thursday.
Finally, if you missed any earlier Parallelograms, make sure you go back and complete them. You can still earn reward points and badges by completing missed Parallelograms.

Cheerio, Simon.