Parallelogram 2 Year 11 17 Sep 2020OK Go

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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 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. MIT 1869 admissions test

2 marks

1.1. This question comes from the 1869 MIT admissions test.

MIT is the Massachusetts Institute of Technology, one of the world’s most prestigious universities.

Can you find the value of x?

Correct Solution: 12

For a full explanation, watch this video.

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

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 In the triangle ABC, AD=BD=CD. What is the size of angle BAC?

  • 60°
  • 75°
  • 90°
  • 120°
  • more information is needed
  • (Not answered)
Show Hint (–1 mark)
–1 mark

Take a look at this video.

A, B, C are all equidistant from D and therefore lie on a circle whose centre is D. BC is a diameter of the circle and BAC is therefore the angle subtended by a diameter at a point on the circumference (the angle in a semicircle).

If you did not know that the angle subtended by a diameter at a point on the circumference is 90 degrees, then take a look at this video.

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

3. Intermediate Maths Challenge Problem (UKMT)

4 marks

3.1 Baldrick can afford to buy either 6 parsnips and 7 turnips or else 8 parsnips and 4 turnips. Both options leave him with no change whatsoever. If, however, he bought only his beloved turnips, how many could he afford?

  • 11
  • 12
  • 13
  • 16
  • 25
  • (Not answered)

Buying two more parsnips and three fewer turnips does not change the total cost and hence two parsnips cost the same as three turnips. Instead of six parsnips, therefore, Baldrick could have bought nine turnips and, together with seven turnips, this makes a total of sixteen turnips. Alternatively, he could have bought twelve turnips instead of the eight parsnips and, together with four turnips, this makes sixteen turnips.

4. OK Go

Enjoy this video for the song “The Writing’s On The Wall” by the band OK Go. It’s full of visual illusions. There is no trick photography!

You would have seen all these effects if you had been in the studio watching with your own eyes.

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

2 marks

4.1 Which word describes many of the illusions (e.g., the steps that look real, but which are actually just painted on the floor)?

  • Anamorphic
  • Anamatronic
  • Anthropic
  • Anthropomorphic
  • Anthologic
  • (Not answered)

5. Intermediate Maths Challenge Problem (UKMT)

5 marks

5.1 Jasmine spends exactly £120 on three types of plants: poisoned ivy, deadly nightshade and triffids. Poisoned ivy plants cost £2 each, deadly nightshade plants cost £9 each, and triffids cost £12 each. She buys twenty plants in total, including at leasst one of each type. How many triffids did she buy?

  • 1
  • 2
  • 3
  • 4
  • more information needed
  • (Not answered)
Show Hint (–1 mark)
–1 mark

You could try some trial and error – it’s slow and boring, but you will eventually get the right answer. If you do get the right answer, then check the solution later, which will give you a more elegant approach.

Show Hint (–2 mark)
–2 mark

Let the number of ivy, nightshade and triffid plants be i, n and t respectively.

Then: 2i+9n+12t=120 and i+n+t=20,
where i>0; n>0; t>0.

Multiplying the second equation by 2 and subtracting the new equation from the first:

7n+10t=80, and thus 7n=108t.

Therefore n is a multiple of 10.

Let the number of ivy, nightshade and triffid plants be i, n and t respectively.

Then: 2i+9n+12t=120 and i+n+t=20,
where i>0; n>0; t>0.

Multiplying the second equation by 2 and subtracting the new equation from the first:

7n+10t=80, and thus 7n=108t.

Therefore n is a multiple of 10, and since 1<n<20, n=10.

Hence 8t=7 and therefore t=1.

I hope you enjoyed the second 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.