Parallelogram 27 Year 10 7 May 2020Alan Turing and the £50 Note

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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.
  • 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. Alan Turing and the £50

In 2019, I was honoured to be invited to join a committee advising the Governor of the Bank of England on which scientist should be placed on the new £50 note to appear in 2021.

The Governor eventually and wisely decided on the mathematician Alan Turing as the face of the £50 note.

3 marks

1.1 If you translate the binary sequence that appears on the draft design of the Turing £50 note, what decimal number do you find? Try working it out by hand, but you can use an online binary-to-decimal convertor if necessary.

Correct Solution: 23,061,912

The binary sequence is 1010111111110010110011000, which is equal to 23,061,912.

2 marks

1.2 What is interesting about the decimal number equal to 1010111111110010110011000?

  • It is Turing’s date of birth
  • It is a random number
  • It is a square number
  • It is a perfect number
  • It is Turing’s Constant of Sentience

The binary sequence is 1010111111110010110011000, which is equal to 23,061,912, or 23/06/1912, which is Turing date of birth.

2 marks

1.3 Alan Turing made contributions to the foundations of mathematics, the theory of computing and artificial intelligence. He is also famous for his contribution to…

  • Astronomy
  • Cryptography
  • Dendrochnology
  • Palaeontology
  • Taxonomy

Alan Turing was a famous cryptographer, helping to break the German Enigma code during the Second World War. He worked at Bletchley Park.

2 marks

1.4 Which of these acronyms involves Turing’s name?

  • BTW
  • CAPTCHA
  • NAFTA
  • OTC
  • ETA
  • EST

CAPTCHA stands for Completely Automated Public Turing Test to tell Computers and Humans Apart.

2. Intermediate Maths Challenge Problem (UKMT)

3 marks

2.1 Professor Rosseforp has an unusual clock. The clock shows the correct time at noon, but the hands move anti-clockwise rather than clockwise.

The clock is very accurate, however, so the hands move at the correct speeds.

If you looked in a mirror at the Professor's clock at 1:30 pm, which of the following would you see?

As the hands move anticlockwise, at 1:30 pm they would have the appearance they would normally have at 10:30.

So, as the clock is viewed in a mirror, it would have the appearance shown in the first image.

3. Intermediate Maths Challenge Problem (UKMT)

4 marks

3.1 The diagram shows two equilateral triangles.

What is the value of x?

  • 70
  • 60
  • 50
  • 40
  • 30
Show Hint (–2 mark)
–2 mark

First, you know that the angles at A, B, C, D, E and F are all 60o. Then you can work out the angles marked with red circles. Then you can work out angle BHE, and from there you can go on to solve the problem.

Show Hint (–1 mark)
–1 mark

Once you know the angle at BHE, then you know the vertical angle, the angle at DHC. You already know that the angle at C is 60o, so you can work out the angle at xo.

Considering the angles at B and E:

∠CBE = (180 - 75 - 60)° = 45°;
∠DEB = (180 - 65 - 60)° = 55°.

Therefore ∠GHB = (45 + 55)° = 100° (exterior angle theorem) and, using the same theorem, ∠HGC = (100 − 60)° = 40°.

4. John Snow

I am writing this Parallelogram at the end of the first week of the UK corona virus lockdown in March 2020. I thought it might be a good time to tell you the story of John Snow, a nineteenth century medical superstar, who invented “epidemiology”, a new mathematical science that has saved millions of lives. Here’s a paragraph about Snow from the website Open Culture.

During the 1854 outbreak of cholera in London, Snow convinced authorities and critics that the disease spread from a contaminated water pump on Broad Street, leading to the now-legendary infographic map below showing the incidences of cholera clustered around the pump. Snow’s persistence resulted in the removal of the handle from the Broad Street pump and has been credited with ending an epidemic that claimed 500 lives. The Broad Street pump map has become “an enduring feature of the folklore of public health and epidemiology," write the authors of an article published in The Lancet.

By finding the location of cholera victims, Snow was able to pinpoint the source of the infection, the water pump, which he was able to disable and thereby halt the spread of the disease. Ever since Snow, doctors and mathematicians have worked hard to use mathematical models to stop the spread of disease. Of course, this is happening right now in order to bring the spread of the corona virus under control.

5. Intermediate Maths Challenge Problem (UKMT)

5 marks

5.1 The figure shows a cube of side 1 on which all twelve face diagonals have been drawn - creating a network with 14 vertices (the original eight corners, plus the six face centres) and 36 edges (the original twelve edges of the cube plus four extra edges on each face).

What is the length of the shortest path along the edges of the network which passes through all 14 vertices?

  • 1+62
  • 4+22
  • 6
  • 8+62
  • 12+122
Show Hint (–3 mark)
–3 mark

If you unwrap the cube, then this is path that you need to follow to pass through every vertex.

Figure (i) shows a net of the cube on which a possible path has been drawn, while figure (ii) shows a diagram of the cube on which the same path has been drawn.

Each edge of the network which joins a corner to a face centre has length, while each edge which joins two adjacent corners has length 12.

So the length of the path shown is 1+12×12, that is 1+62. This is the length of the shortest path along the edges of the network which passes through all 14 vertices.

To prove this, we first note that to connect the 14 vertices we need a minimum of 13 edges, so the length of the shortest path must be at least 13×12.

A path of this length would move alternately between corners and face centres, but as there are 8 corners and 6 face centres this is impossible. At least one of the edges on the shortest path, therefore, must join two corners. So the length of the shortest path must be at least 1+12×12.

The diagrams show that such a path does exist so we are able to conclude that is indeed the length of the shortest path.

6. Word riddles

Just a little something (i.e., no marks) that you might like from the website of Professor Richard Wiseman. These are word riddles.

For example, the solution to the first one is “Just between you and me”, because the word JUST is between YOU and ME.

0 marks
  1. YOU JUST ME

  2. STAND
    I

  3. BRO KEN

  4. ¦ R ¦ E ¦ A ¦ D ¦ I ¦ N ¦ G ¦

  5. TIMING TI-MING

  1. Just between you and me
  2. I understand
  3. Broken in two
  4. Reading between the lines
  5. Split second timing

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.