Parallelogram 2 Year 8 19 Sep 2019Jurassic Park mathematics

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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.

Welcome to the second of our Parallelograms designed for Year 8 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.
  • 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 look at the solution sheet to see where you went wrong, and then next time you’ll know what to do.

1. More on the number 6,174

Last week we looked at 6,174, the Kaprekar constant. If you have not completed last week’s Parallelogram, then take a look at it after you complete this week’s list of questions. Remember, you can still earn reward points on a past Parallelogram.

As well as being the Kaprekar constant for 4-digit numbers, the number 6,174 has a few interesting qualities. For example, it equals (181 + 182 + 183), which is (18 + 324 + 5,832).

6,174 is also known as a harshad number, because it is divisible by the sum of its digits:

6,174 = (6 + 1 + 7 + 4) x 343.

The number 18 is another harshad number, because it is also divisible by the sum of its digits:

18 = (1 + 8) x 2.

By the way the word "harshad" comes from the ancient Indian Sanskrit language and it means joy-giver. So, I hope they are giving you at least a tiny bit of joy.

1 mark

1.1. Is the number 42 a harshad number?

  • Yes, it’s a harshad number
  • No, it’s not

42 is a harshad number, because (4 + 2 = 6) … and 6 divides into 42.

1 mark

1.2. Is the number 44 a harshad number?

  • Yes, it’s a harshad number
  • No, it’s not

44 is NOT a harshad number, because (4 + 4 = 8) … and 8 will not divide into 44.

1 mark

1.3. Is the number 198 a harshad number?

  • Yes, it’s a harshad number
  • No, it’s not

198 is a harshad number, because (1 + 9 + 8 = 18) … and 18 will divide into 198.

After all, 11 x 18 = 198.

3 marks

1.4. 152 and 153 are neighbouring harshad numbers. If you look at bigger numbers, what is the next harshad number?

Correct Solution: 156

154 and 155 are not harshad numbers, but 156 is a harshad number, because (1 + 5 + 6 = 12) … and 12 will divide into 156.

2. Jurassic Park mathematics

Jurassic Park is one of Steven Spielberg’s most famous films, and it is based on the idea that biotechnology could use preserved DNA to bring dinosaurs back to life. However, one of its main characters is a mathematician, Dr Ian Malcolm (played by Jeff Goldblum), who specializes in a branch of mathematics known as "Chaos Theory". He tries to explain chaos theory in this clip (but he does not bother discussing the mathematical equations behind chaos theory).

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

3. Junior Maths Challenge Problem (UKMT)

If you are a Year 8 student, then it 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 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 A swimming club has three categories of members: junior, senior, veteran. The ratio of junior to senior members is 3:2 and the ratio of senior members to veterans is 5:2.

Which of the following could be the total number of members in the swimming club?

Show Hint (–1 mark)
–1 mark

The 5:2 ratio suggests that we might have 5 seniors and 2 veterans, but if we have only 5 seniors then the 3:2 ratio would mean 7.5 juniors, which is clearly impossible. So, instead of 5 seniors and 2 veterans, what other numbers would agree with a 5:2 ratio?

  • 30
  • 35
  • 48
  • 58
  • 60

The 5:2 ratio suggests that we might have 5 seniors and 2 veterans, but if we have only 5 seniors then the 3:2 ratio would mean 7.5 juniors, which is clearly impossible.

However, the 5:2 ratio also suggests that we might have 10 seniors and 4 veterans, and if we have only 10 seniors then the 3:2 ratio would mean 15 juniors, which is a much more sensible answer, with a total of 10 + 4 + 15 = 29 members. But this answer is not one of the choices, so we need to look for a multiple of 29, which is 58.

If you missed the first Parallelogram, then try to go back and complete it. After all, you can earn reward points and badges by completing each Parallelogram. Find out more by visiting the Rewards Page after you hit the SUBMIT button.

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.