Parallelogram 34 Year 10 2 Jul 2020Revolutionary Riddles

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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.
  • 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. Lockdown maths

I am writing this Parallelogram during lockdown (May 2020), and I recently saw this neat little trick.

Instructions

  1. Choose a number between 1 and 9
  2. Multiply it by 3
  3. Add 3
  4. Multiply by 3 again
  5. Add the two digits of the number you get together
  6. The number you're left with is where you will be travelling to
1 mark

1.1 Can you calculate your next travel destination?

  • 1. Singapore
  • 2. Spain
  • 3. India
  • 4. Thailand
  • 5. Malaysia
  • 6. Indonesia
  • 7. Brazil
  • 8. England
  • 9. Stay at home
  • 10. Australia
  • 11. Cambodia
  • 12. Vietnam
  • 13. Japan
  • 14. South Korea
  • 15. Iceland
  • 16. Canada
  • 17. Mexico
  • 18. New Zealand

If you pick x, then you will arrive at the number 33x+3, or 9x+9, or 9x+1. So the final number is a multiple of 9, and the digits of a multiple of 9 add up to 9... so you end up staying at home.

2. Intermediate Maths Challenge Problem (UKMT)

3 marks

2.1 Imogen used her calculator and multiplied a number by 20 instead of by 2. What could she now do to obtain the correct answer?

  • Divide by 20
  • Divide by 40
  • Multiply by 10
  • Multiply by 0.5
  • Multiply by 0.1

Imogen's result is ten times bigger than it should be, and therefore dividing by 10 will correct her mistake. Multiplying by 0.1 is equivalent to dividing by 10.

3. Intermediate Maths Challenge Problem (UKMT)

4 marks

3.1 Which of the following statements is false?

  • An octagon has twenty diagonals.
  • A hexagon has nine diagonals.
  • A hexagon has four more diagonals than a pentagon.
  • A pentagon has the same number of diagonals as it has sides.
  • A quadrilateral has twice as many diagonals as it has sides.

The number of diagonals in an n-sided polygon is 12nn3. This can be used to show that the first four statements are all correct. A quadrilateral has half as many diagonals as sides, not twice as many.

4. Revolutionary Riddles

Take a look at this great video about riddles relating to revolving or rotating objects. It poses 4 mysteries, which will all be answered in the answer section (after you hit the SUBMIT button). And have a go at answering the question below, which relates to one of the mysteries.

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

3 marks

4.1 If you gently jog one lap at a speed of 10 km/h, how fast would you have to run the second lap so that the average speed for each laps is 20 km/h?

  • 5 km/h
  • 10 km/h
  • 20 km/h
  • 100 km/h
  • It cannot be done

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

5. Intermediate Maths Challenge Problem (UKMT)

5 marks

5.1 The Queen of Hearts has lost her tarts! She is sure that those knaves who have not eaten tarts will tell her the truth and that the guilty knaves will tell lies. When questioned, the five knaves declare:

  • K1: "Only one of us ate them"
  • K2: "Only two of us ate them"
  • K3: "Only three of us ate them"
  • K4: "Only four of us ate them"
  • K5: "All five of us ate them"

How many of the knaves ate the tarts?

  • 1
  • 2
  • 3
  • 4
  • 5
Show Hint (–1 mark)
–1 mark

The five knaves all say that a different number ate the tarts, which means that only one of them can be telling the truth.

The five knaves all say that a different number ate the tarts, which means that only one of them can be telling the truth.

Only honest knaves who have not eaten any tarts will tell the Queen the truth, so that means that only one knave out of the five did not eat the tarts.

K4 is the only honest knave who didn't eat the tarts, so that means K1, K2, K3 and K5 at the tarts.

6. Thanks, and how long are all the fish?

2 marks

6.1 Fred went fishing. He caught a fish that had a length of 30 cm plus half its own length.

How long was the fish?

Correct Solution: 60 cm

If L = Length of fish, then

L=30+12L
12L=30
L=60 cm

2 marks

6.2 Fred went fishing again. This time he claimed that he caught a fish that had a length of 30 cm plus double its own length. Nobody saw the fish. Maybe he was telling a lie. If fish really was (30 cm plus double its own length), then what would this mean for the length of the fish? Don’t worry if your answer seems impossible.

Correct Solution: -30 cm

If L = Length of fish, then

L=30+2L
L=30
L=30 cm

There will be more next week, 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.