Parallelogram 13 Level 6 30 Nov 2023Gravity

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Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

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

These challenges are a random walk through the mysteries of mathematics, most of which will be nothing to do with what you are doing at the moment in your classroom. Be prepared to encounter all sorts of weird ideas, including a few questions that appear to have nothing to do with mathematics at all.

  • Tackle each Parallelogram in one go. Don’t get distracted.
  • When you finish, remember to hit the SUBMIT button.
  • Finish by Sunday night if your whole class is doing parallelograms.

IMPORTANT – it does not really matter what score you get, because the main thing is that you think hard about the problems... and then examine the solution sheet to learn from your mistakes.

1. Intermediate Maths Challenge Problem (UKMT)

3 marks

1.1 2001 = 3 × 23 × 29. Which of the following numbers is also the product of exactly three distinct prime numbers?

  • 45
  • 60
  • 91
  • 105
  • 330
  • (Not answered)

105 = 3 × 5 × 7.

2. The Gravity of the Situation

Some physics brought to you by YouTube channel, minutephysics. Watch carefully.

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

1 mark

2.1 Which of these objects does gravity not attract?

  • Light
  • Dogs
  • Guitars
  • Hopes and fears
  • (Not answered)
3 marks

2.2 Isaac Newton discovered that the strength of the gravitational force exerted by two objects on each other decreases by the square of the distance between them.

Let’s say the gravitational force between objects A and B is 81 N. What is the gravitational force if they move three times further away from each other?

Correct Solution: 9 N

Show Hint (–1 mark)
–1 mark

By moving three times further away, the force will decrease by a factor of 32=9.

By moving three times further away, the force will decrease by a factor of 32=9.


3. Intermediate Maths Challenge Problem (UKMT)

4 marks

3.1 A square patchwork quilt is made by joining four square pieces of cloth, with each piece coloured grey and white as shown.

Only edges of the same colour are sewn together.

How many different quilt patterns are possible?

Two patterns are considered to be the same if one can be rotated to look exactly like the other.

  • 3
  • 4
  • 5
  • 6
  • 8
  • (Not answered)

The only possible distinct patterns are:

4. Maths Meme

Have you ever seen someone on social media share a ‘difficult’ maths problem which can be solved easily by following the correct order of operations? They usually go like this:

“Only 10% of people will get this right! What is 3 + 2 × 4?”

They are usually quite mundane, but here’s one which is genuinely interesting.

230 – 220 × 0.5 = ?

You can easily see the incorrect answer of 5 if you perform the subtraction before the multiplication, and you can calculate the correct answer by subtracting half of 220 from 230. But actually the final statement is true! See if you can spot why, before answering the question below which gives it away.

4 marks

4.1 Work out the value of 230 – 220 × 0.5, and subtract from it the answer to 5 factorial (5 × 4 × 3 × 2 × 1, which is denoted by 5!). What is your answer?

Correct Solution: 0

The answer is 0 because 230 – 220 × 0.5 = 5! (That is, 5 factorial), just like the final line stated! Get it?

5. Intermediate Maths Challenge Problem (UKMT)

5 marks

5.1 The diagram shows an irregular hexagon with interior angles all equal to 120° made by cutting the corners off a piece of card in the shape of an equilateral triangle with sides of length 20 units.

An identical hexagon could also be made by cutting the corners off a different equilateral triangle: what is the side length of this triangle?

  • 23
  • 25
  • 27
  • 29
  • 31
  • (Not answered)
Show Hint (–2 mark)
–2 mark

In the construction in the question, the sides with length 13, 11 and 10 are segments of the sides of the original equilateral triangle, and the sides with length 3, 4 and 6 are the ‘cuts’.

Can you imagine an equilateral triangle in which these roles are reversed, i.e. 3, 4 and 6 are segments of the sides of the equilateral triangle, and 13, 11 and 10 are the ‘cuts’?

Show Hint (–1 mark)
–1 mark

By extending the short sides, one can construct another triangle.

Since all the angles involved are 60°, the new small triangles and the new large triangle are all equilateral.

By extending the short sides, one can construct another triangle.

Since all the angles involved are 60°, the new small triangles and the new large triangle are all equilateral.

The sides of the new, large triangle are 13 + 3 + 11 = 11 + 6 + 10 = 10 + 4 + 13 = 27.

I hope you enjoyed this Parallelogram. There will be more next week, and the week after, and the week after that. So check your email or return to the website on Thursday at 3pm.

In the meantime, you can find out your score, the answers and go through the answer sheet as soon as you hit the SUBMIT button below.

When you see your % score, this will also be your reward score. As you collect more and more points, you will collect more and more badges. Find out more by visiting the Rewards Page after you hit the SUBMIT button.

It is really important that you go through the solution sheet. Seriously important. What you got right is much less important than what you got wrong, because where you went wrong provides you with an opportunity to learn something new.

Cheerio, Simon.