PG 27 2 May 2019Circle Mystery

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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.
  • Tackle each Parallelogram in one go. Don’t get distracted.
  • Finish by midnight on Sunday if your whole class is doing parallelograms.
  • Your score & answer sheet will appear immediately after you hit SUBMIT.
  • Don’t worry if you score less than 50%, because it means you will learn something new when you check the solutions.

1. Intermediate Maths Challenge Problem (UKMT)

2 marks

1.1 What is 25% of 34?

  • 316
  • 14
  • 13
  • 1
  • 3

As 25% of a number is one quarter of it, 25% of 34 is 14×34=316.

2. Intermediate Maths Challenge Problem (UKMT)

3 marks

2.1 To the nearest whole number, what percentage of 34 is 23?

Correct Solution: 89 %

We can re-write the question as:

23=x100×34

Therefore,

x=4×100×23×3=88.9%

3. Circle mystery

This is a video from the terrific Numberphile channel on YouTube. It features one of the world’s greatest mathematicians, Neil Sloane. As well making breakthroughs in mathematics, he has written two books on rock-climbing. Watch this video, which looks at an apparently simple question about circles … then answer the questions after.

2 marks

3.1 According to the rules in the video, which of these arrangements of two circles is not allowed?

  • A
  • B
  • C
  • D
  • E
2 marks

3.2 According the rules, how many ways can you arrange two circles?

Correct Solution: 3

2 marks

3.3 Here are some arrangements for three circles. According the rules, which two arrangements are the same? Add up the two numbers of the duplicate arrangements and enter it in the box.

Correct Solution: 16

Arrangements 7 and 9 are the same, so the total is 16.

3 marks

3.4 According the rules, how many ways can you arrange four circles?

Correct Solution: 173

3 marks

3.5 According the rules, how many ways can you arrange five circles?

Correct Solution: 16951

4. Intermediate Maths Challenge Problem (UKMT)

3 marks

4.1. Zac halves a certain number and then adds 8 to the result. He finds that he obtains the same answer if he doubles his original number and then subtracts 8 from the result.

What is Zac’s original number?

  • 823
  • 913
  • 923
  • 1013
  • 1023

Let Zac’s original number be x. When Zac halves this and adds 8 he obtains 12x+8. When he doubles his original number and subtracts 8, the result is 2x8.

As these answers are the same, 12x+8=2x8. Therefore 3/2 x = 16. Hence x=23×16=323=1023.

Before you hit the SUBMIT button, here are some quick reminders:

  • You will receive your score immediately, and collect your reward points.
  • You might earn a new badge... if not, then maybe next week.
  • Make sure you go through the solution sheet – it is massively important.
  • A score of less than 50% is ok – it means you can learn lots from your mistakes.
  • The next Parallelogram is next week, at 3pm on Thursday.
  • Finally, if you missed any earlier Parallelograms, make sure you go back and complete them. You can still earn reward points and badges by completing missed Parallelograms.

Cheerio, Simon.