Parallelogram 32 Year 10 18 Jun 2020Lewis Carroll's Pillow Problem

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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.
  • 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. Lewis Carroll's Pillow Problem

Just watch and enjoy this video by the Alex Bellos (author of some great books about mathematics (and football)) for the Numberphile YouTube channel. No questions, just a video.

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

2. Intermediate Maths Challenge Problem (UKMT)

3 marks

2.1. A sheet of A4 paper (297 mm × 210 mm) is folded once and then laid flat on the table.

Which of these shapes could not be made?

  • A
  • B
  • C
  • D
  • E

For B to be possible, the diagonal of a rectangle would need to be an axis of symmetry.

3. Intermediate Maths Challenge Problem (UKMT)

4 marks

3.1 In a right angled triangle the two shorter sides have lengths 10cm and 5cm. Which of the following approximations is closest to the length of the hypotenuse?

  • 11cm
  • 11.5cm
  • 12cm
  • 12.5cm
  • 13cm

Let the hypotenuse be of length h cm. Then h2=100+25=125.

Therefore h=125; 112=121 and 122=144, therefore 11 < h < 12.

11.52=11+122=112+2×11×12+14=13214, therefore 11 < h < 11.5.

11.252=11+142=121+2×11×14+116, therefore 11.252>126.5, therefore 11 < h < 11.25.

The length of the the hypotenuse is therefore closer to 11 cm than 11.5 cm.

4. An interesting tip

I spotted this on the internet, a bill from a restaurant, with a very interesting tip, but the tip has been covered with a red patch.

4 marks

4.1 What is the tip?

  • Something to do with squares
  • Something to do with triangles
  • Something to do with circles
  • Something to do with cubes
  • Something to do with hypercubes

If you subtract the amount from the total, then you find that the tip is 3.14... which is the common approximation for π, which is “something to do with circles”.

5. Intermediate Maths Challenge Problem (UKMT)

5 marks

5.1 In the Soft Boulder Café each table has 3 legs, each chair has 4 legs and all the customers and the three members of staff have two legs each. There are four chairs at each table. At a certain time, three-quarters of the chairs are occupied by customers and there are 206 legs altogether in the café. How many chairs does the café have?

  • 20
  • 24
  • 28
  • 32
  • 36

206 - the 6 legs of the staff = 200. On average, at each table, there are three table legs, 16 chair legs, and 6 customer legs, which gives a total of 25 legs per table. 200 ÷ 25 = 8, hence there are 8 tables and therefore 32 chairs.

6. Optical illusions

Here is a great video from YouTuber Zach King, featuring optical illusions about furniture. They are all real, except the anamorphic chair becoming real must be a camera trick.

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

2 marks

6.1 Ashley HomeStore provided a grey chair, which appears to float after the legs are removed. How?

  • smoke
  • mirrors
  • micro silk suspension
  • mini-jets
  • supernatural forces

There will be more next week, 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.