Week 1School of Rock

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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.

Welcome to the first of our Parallelograms designed for Year 7 students, a collection of mathematical challenges designed to stretch your brain and make your neurons more squiggly.

These challenges are a random walk through the mysteries of mathematics. Be prepared to encounter all sorts of weird ideas, including some questions that have nothing to do with mathematics.

  • Tackle each Parallelogram in one go. Don’t get distracted.
  • When you finish, remember to hit the SUBMIT button.
  • NEW! You can new earn reward points and badges – find out more here.
  • Finish by midnight on Sunday 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. Happy Numbers

This is a video from the hugely successful maths video channel Numberphile. The channel has received over 300 million views, so I would encourage you to explore some of its other videos if you want to learn more about maths.

Ria Symonds from the University of Nottingham explains why 7 is a happy number. Watch it and answer the questions below.

So for a number to be happy, just take the digits, square each digit and add all the squares to create a new number. Then repeat the process with the new number, and continue until you end up with the number 1 or find the numbers stuck in a repeating loop that does not contain 1.

If you end up with the number 1, then the number is happy. For example, 19 is happy, because it has the digits 1 and 9:

12 + 92 = 82
82 + 22 = 68
62 + 82 = 100
12 + 02 + 02 = 1.

Try to work out if the following numbers are happy or unhappy.

1 mark

1.1. Is 100 a happy number?

  • Happy
  • Sad

Of course 100 is happy, because 12 + 02 + 02 = 1.

2 marks

1.2. Is 888 a happy number?

  • Happy
  • Sad

888 is happy, because 82 + 82 + 82 = 192... Then this turns into 86.... Then this turns into 100... Then this turns into 1.

2 marks

1.3. Is 145 a happy number?

  • Happy
  • Sad

145 is a sad number, because it leads to the chain 145 → 42 → 20 → 4 → 16 → 37 → 58 → 89 → 145.

We can stop there, because we started with 145, so we have created a loop. The numbers will go round and round and never reach 1.

2 marks

1.4. Is 5,555 a happy number?

  • Happy
  • Sad

5,555 is happy, because 52 + 52 + 52 + 52 = 25 + 25 + 25 + 25 = 100.

100 is then transformed into 12 + 02 + 02 = 1.

2. School of Rock

I recently watched “School of Rock”, in which Jack Black plays an unemployed musician who pretends to be a teacher. This clip shows Black’s attempt to couple of his love of music with his shaky grasp of maths.

2 marks

2.1 The teacher ask what is six multiplied by a billion, but what is a billion times a billion called? You might need to google this.

  • Gazillion
  • Trillion
  • Quadrillion
  • Quintillion
  • Sextillion

If you google “large numbers” later, then you will learn everything you need to know about the names of large numbers. In short, however, a billion is 109, so a billion times a billion is 1018, which is known as a quintillion.

3. Junior Maths Challenge Problem

If you are a Year 7 student, then it is likely that you will be taking part in the United Kingdom Maths Trust (UKMT) competition known as the Junior Maths Challenge (JMC). 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 7 and Year 8 (!) 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 Which of the following expressions has the largest value?

  • 1 − 0.1
  • 1 − 0.01
  • 1 − 0.001
  • 1 − 0.0001
  • 1 − 0.00001

In each case a number is subtracted from 1. The smaller the number we subtract, the larger the answer will be. So the largest value is obtained when the smallest number is subtracted.

I hope you enjoyed the first Parallelogram of the year. 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.