Parallelogram 22 Year 8 12 Mar 2020Rattlebacks

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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. Musical Challenge

1 mark

1.1. This clip is from University Challenge on BBC2 – listen carefully and add up all the numbers in the song.

Correct Solution: 176

M62 + 39 miles + 45 minutes + 30 miles = 176.

2. Junior Maths Challenge Problem (UKMT)

1 mark

2.1 A pattern that repeats every six symbols starts as shown below:

♥ ♣ ♥ ♦ ♥ ♠ ♥ ♣ ♥ ♦ ♥ ♠ ...

Which is the 1001st symbol?

Our pattern repeats after six symbols, so to find the 1,001st symbol we need to work out: 1,001 ÷ 6 = 166 remainder 5.

So, the 6-symbol pattern repeats itself 166 times, and the 5th symbol will sit in the 1,001st position. So the answer is the heart symbol, ♥.

3. Junior Maths Challenge Problem (UKMT)

3 marks

3.1. One day Beth, Carolyn and George read a book of 240 pages. They take it in turns to read a page, always in the order Beth, Carolyn and George.

The next day their cousin Sam comes to stay and they read the book again, taking it in turns to read a page, always in the order Sam, George, Carolyn and Beth.

How many pages are read by the same person on both days?

Correct Solution: 60

Beth used to read every 3rd page 3n+1, ie 1, 4, 7, 10, 13, 16, 19, 22, 25, 28,...

Now Beth reads every 4th page, ie 4, 8, 12, 16, 20, 24, 28,...

Beth read 1 in 3 of the original pages, and then Beth reads 1 in 4 of them again.

So, if she read 2403=80 pages originally, she now reads 20 of those same pages again.

The same will be true for Carolyn and George, so 3 × 20 pages = 60 pages will be read by the same person when the book is re-read with the help of Sam.

4. How does the sun shine?

Derek Muller from YouTube’s Veritasium channel asks people to explain the mechanism that is responsible for the sun shining. Have a think yourself before you watch Derek explain the physics behind sunshine.

5. Junior Maths Challenge UKMT bonus

As the Junior Maths Challenge is now not far away, here is a bonus (and particularly tough) JMC question to stretch your brain.

3 marks

5.1. After playing 500 games, my success rate at Spider Solitaire is 49%. Assume that I now lose every third game, so that after 500 games my results are Win, Win, Loss, Win, Win, Loss, … .

What is the least number of extra games I need to play in order that my success rate becomes at least 50%?

Correct Solution: 26

Since I have won 49% of my first 500 games, so far I have won:

49100×500=49×5=245 games.

So I have lost 500245=255 games. I need now to win enough games so that I have won as many as I have lost.

So, assuming I win 2 out of 3 games from now on, I get the following pattern:

# games Wins Losses
500 245 255
503 247 256
506 249 257
509 251 258
512 253 259
515 255 260
518 257 261
521 259 262
524 261 263
527 263 264
530 265 265

So, it would be easy to assume that I need to play 30 more games, so that the number of wins equals the number of losses. However, let’s take a closer look at what happens towards the end, bearing in mind the win-win-loss pattern.

# games Wins Losses
524 261 263
525 262 263
526 263 263
527 263 264
528 264 264
529 265 264
530 265 265

So, the number of wins equals the number of losses three times – after 526, 528 and 530 games. Therefore, the earliest that the number of wins equals the number of loses is after 526 games, or after 26 more games.

6. Rattleback

Watch this video of one of the most mysterious objects in the universe. It’s called a rattleback, and you can win one if you succeed at the challenge in the Additional Stuff section below.

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.

Additional Stuff

VAX! – a game about epidemic prevention.

Take a look at the game on the VAX! website, which is all about showing how maths can help reduce the spread of disease. We are all connected to each other and it is through these connections that diseases spread. You can reduce the spread of disease by breaking these connections, either by vaccinating people or putting them in quarantine.

It is definitely worth looking at the whole site and learning about the maths of disease, networks and connections, but if you want to jump straight in to the game then this is how to find it and how to win a rattleback.

  • Go to http://vax.herokuapp.com/game
  • Click on EASY (try MEDIUM & HARD later)
  • Start vaccinating those dots that are most likely to spread disease.
  • When the disease starts, begin quarantining dots.
  • If you want to try again, hit RETRY after you obtain your score.

If you would like to win a rattleback, then please send us your best VAX! score. The top three scores received by 12pm on Tuesday 19th March will win a rattleback. Just email me at prizes@parallel.org.uk and include an image of your bar chart score. Good luck.