Parallelogram 8 Level 6 26 Oct 2023Colliding Vortexes

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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.

These challenges are a random walk through the mysteries of mathematics, most of which will be nothing to do with what you are doing 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 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. Intermediate Maths Challenge Problem (UKMT)

3 marks

1.1 A tourist attraction which opens every day needs 30,000 visitors per day on average to break even. Last week there were 120,000 visitors. What is the number of visitors needed this week to break even over the two-week period?

  • 43,000
  • 90,000
  • 180,000
  • 210,000
  • 300,000
  • (Not answered)

Over a two-week period, the required number of visitors is 14 × 30,000 = 420,000. Hence the number required this week is 420,000 - 120,000 = 300,000.

2. Colliding vortexes

Science can be very beautiful, and it can also be hard work. This video from YouTube channel SmarterEveryDay, on colliding vortexes (or vortices, if you prefer), shows this brilliantly.

By the way, a vortex is a mass of gas or liquid with a whirling or circular motion, such as a smoke ring.

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

3 marks

2.1 When Destin completed his experiment, he realised that his exploration was not about fluid dynamics, but rather about…

  • Mechanics and dynamics
  • Vacuums and pressure
  • Persistence and patience
  • Accuracy and detail
  • Tom and Jerry
  • (Not answered)

3. Prime numbers

Number theorists love to find formulas which generate prime numbers. You may have heard of Mersenne’s formula 2p1, where p is a prime number, which has generated the 18 largest known primes.

Here is a simpler procedure which seems to generate prime numbers. Write a list of 3s, and end with the digit 1. For example, 31 is prime, so is 331, and so is 3,331.

2 marks

3.1 What is the smallest number of this type (lots of 3s ending with a 1) which is not prime?

Correct Solution: 333333331

Show Hint (–2 mark)
–2 mark

You may have a ‘factorise’ function on your calculator which could help here. On the Casio fx-83GT X, it is the second function of the button for degrees/minutes/seconds (o ’ ”).

If not, you can use and replace the number at the end of the URL.

4. Intermediate Maths Challenge Problem (UKMT)

3 marks

4.1 A wire in the shape of an equilateral triangle with sides of length 9 cm is placed on a flat piece of paper.

A pencil is held in the hole at the centre of a disc of radius 1cm, and the disc is rolled all the way around the outside of the wire, and then all the way around the inside of the wire.

What shape is drawn by the pencil?

  • A
  • B
  • C
  • D
  • E
  • (Not answered)

On the outside of the wire, the pencil describes an arc of a circle as the disc rolls around each of the corners of the triangle, but this does not happen when the disc moves around the inside of the wire.

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.