Week 4Strange Breakfasts

This is a preview of Parallel. You have to login or create an account, to be able to answer questions and submit answers.

Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

  1. a portmanteaux 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.
  • When you finish, remember to hit the SUBMIT button.
  • Finish by midnight on Sunday if your whole class is doing parallelograms.
  • Make sure you check the solution sheet, celebrate your successes and (most important of all) learn from your mistakes.

1. Grandi’s series

Guido Grandi (1671 – 1742) was an Italian monk, philosopher, mathematician and engineer. In 1703-ish, he wondered what happens if you add +1 and −1 over and over again, forever.

1 – 1 + 1 – 1 + 1 – 1 + 1 – 1 + 1… = ?

James Grime, a presenter on the terrific Numberphile YouTube channel, explains how you might answer this question. But, before you watch it, spend 30 seconds figuring out what you think the answer might be.

2 marks

1.1. There is no question associated with this video, but instead I want you to try and explain Grandi’s series to a parent or an older sibling and see if you can explain how a mathematical series can add up to three completely different answers. I am assuming that you will do this. Can I trust you?

  • Yes
  • No

2. The most important meal of the day

I recently came across this video that shows how there is fascinating science (and therefore mathematics) all around us. Just look closely at what’s staring you in the face and you will see how wonderful and curious the world can be.

If you want to see this for yourself, ask your science teacher to try it out. Your teacher will need a strong magnet (probably neodymium). One of the best cereals to do this with is Kellogg’s Bran Flakes, which contains 11.6mmg of iron per 100g of cereal.

3 marks

2.1. If you took all the bowls of cereal that were on British breakfast tables this morning and tipped them on to the pitch at Wembley Stadium (before they were eaten), how deep would the pool of cereal be?

Imagine that there was wall built around the pitch, so that the wall and the pitch form a giant cereal dish.

You will need to estimate a few things in order to decide the right answer.

  • 1 m
  • 10 m
  • 100 m
  • 1,000 m
  • 10,000 m
Show Hint (–1 mark)
–1 mark

I estimated that the average cereal was 7 cm x 7 cm x 5 cm, assuming a square dish.

The key numbers are the volume of a bowl of cereal, the number of bowls of cereal across the country, and the size of Wembley Stadium.

I reckon that a bowl of cereal might be 7 cm x 7 cm x 5 cm ≈ 250 cm3.

There are 70 million people in the UK, and maybe 1 in 3 have cereal for breakfast ≈ 23,000,000 bowls.

Total volume of cereal ≈ 23,000,000 x 250 cm3 ≈ 6,000,000,000 cm3 ≈ 6,000 m3.

The pitch at Wembley is roughly 120 m x 60 m = 7,000 m2.

This means that the pool of cereals in Wembley Stadium would be roughly 6,000/,7000 ≈ 1 m deep.

Obviously, this is a very rough calculation, with lots of estimations along the way, and I have rounded up and rounded down each step. For example, I later googled "Wembley pitch", and it is 105 m × 68 m, which is smaller than I estimated. The idea is that sometimes you will overestimate and sometimes you will underestimate and these errors will generally partly cancel out. I don’t know the actual true answer, but I strongly suspect that it is between five times less and five times more than my answer.

3. Intermediate Maths Challenge Problem

As I mentioned in previous Parallelograms, if you are a Year 9 student, then it is likely that you will be taking part in the United Kingdom Maths Trust (UKMT) competition known as the Intermediate Maths Challenge (IMC). If you do particularly well, you might earn yourself a gold, silver or bronze certificate, but you will have to work hard as you will be competing against Year 9, Year 10 and Year 11 students from across the country.

Your teachers will help you prepare for this national maths competition, but in each week's Parallelogram we will always include one UKMT Junior Maths Challenge question.

3 marks

3.1 The angles of a triangle are in the ratio 2:3:5. What is the difference between the largest angle and the smallest angle?

  • 18°
  • 36°
  • 45°
  • 54°

The sum of the angles in a triangle is 180°. Since the angles are in the ratio 2:3:5 we have to divide 180° in the ratio 2:3:5.

The difference between the largest angle the smallest angle is therefore
522+3+5 ×180° = 310×180° = 54°

If you missed the earlier Parallelograms, then try to go back and complete them. After all, you can earn reward points and badges by completing each Parallelogram. Find out more by visiting the Rewards Page … after you have hit the SUBMIT button, gone through the answer sheet and seen your score.

There will be another Parallelogram 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.

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.