Parallelogram 1 Year 11 31 Aug 2020Prince Rupert’s Drops

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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 first of our Parallelograms designed for Year 11 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, most of which will be nothing to do with what you are studying at the moment in your classroom. Be prepared to encounter all sorts of weird ideas, including a few questions that appear to have nothing to do with mathematics at all.

  • 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. Blue Plaque

2 marks

1.1. This is an alternative heritage blue plaque and a photo of the mathematician whose life is remembered.

What is the name of the mathematician? Just type in the surname.

Correct Solution: Venn

2. Intermediate Maths Challenge Problem (UKMT)

Every Parallelogram will contain two or more UKMT Intermediate Maths Challenge problems. This first one is fairly simple, but others will be much trickier... and for all of them, remember, NO CALCULATORS!

3 marks

2.1 Given that x and y are positive whole numbers and x2+2=y3, which of the following is a possible value of x?

  • 2
  • 3
  • 4
  • 5
  • 6


This is in fact the only solution of this equation for which x and y are positive whole numbers. Another way of looking at this is to say that 26 is the only whole number which is 'sandwiched' between a perfect square and a perfect cube. This was first proved by the French mathematician, Pierre de Fermat, in the 17th century.

3. Intermediate Maths Challenge Problem (UKMT)

4 marks

3.1 The pattern 123451234512345... is continued to form a 2000-digit number. What is the sum of all 2000 digits?

  • 6000
  • 7500
  • 30,000
  • 60,000
  • 75,000

The number may be divided up into 400 blocks of '12345'. The sum of the digits in each block is 15 and hence the sum of all 2000 digits is 400 × 15 = 6000.

Alternatively, the mean of the digits which make up the number is 3, and therefore the sum of the digits is 2000 × 3 = 6000.

4. Prince Rupert’s drops

This video is from one of my favourite YouTube channels, “Smarter Every Day”. I hope you will watch some of the other videos made by Destin Sandlin.

Take a look at this video about Prince Rupert’s drops, and answer the question below.

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

2 marks

4.1 Destin explains the strength of a Prince Rupert’s drop by comparing it to…

  • A bicep
  • A spring
  • A catapult
  • A arch bridge
  • An elephant

5. Intermediate Maths Challenge Problem (UKMT)

5 marks

5.1 The large circles in each figure have the same radius, R.

Which shaded area is the greatest?

  • A
  • B
  • C
  • D
  • E

Note: as this question is worth 5 marks, using all 5 hints will mean you will score 0 marks.

Show Hint (–1 mark)
–1 mark

Area A =2×π12R2=12πR21.6R2.

Show Hint (–1 mark)
–1 mark

Area B =3×12×R×Rsin120°=343R21.3R2.

Show Hint (–1 mark)
–1 mark

Area C =2R2πR2=4πR20.9R2.

Show Hint (–1 mark)
–1 mark

Area D =2×12×R2=R2.

Show Hint (–1 mark)
–1 mark

Area E =πR22R2=π2R21.1R2.

Area A =2×π12R2=12πR21.6R2.

Area B =3×12×R×Rsin120°=343R21.3R2.

Area C =2R2πR2=4πR20.9R2.

Area D =2×12×R2=R2.

Area E =πR22R2=π2R21.1R2.

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.