Parallelogram 20 Year 9 27 Feb 2020Four-Colour Map Theorem

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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. Four-colour map theorem (part 1)

This week’s Parallelogram is a bit different. You are going to watch a video by James Grime about the four-colour map theorem, one of the most famous problems in the history of mathematics. The video is split into several parts, and there are question after each video clip. Good luck.

1 mark

1.1. Roughly how long did it take to prove the 4 colour map theorem?

  • A day
  • A week
  • A year
  • A century
  • Not yet solved
2 marks

1.2. This simple map needs two colours:

What is the minimum number of colours needed to colour this map?

Correct Solution: 2

2 marks

1.3. What is the minimum number of colours needed to colour this map?

Correct Solution: 3

2 marks

1.4. What is the minimum number of colours needed to colour this map?

Correct Solution: 3

2 marks

1.5. What is the minimum number of colours needed to colour this map?

Correct Solution: 4

2. Four-colour map theorem (part 2)

In this clip, James go on to explain how we can think about maps in terms of networks. It seems odd, but it actually makes life easier.

2 marks

2.1 Which of the following networks is the odd one out?

  • A
  • B
  • C
  • D
  • E

All the networks have four nodes/blobs. All the networks have every node connected to 3 lines. However, network D has 6 lines, but all the others have 5 lines. If you remake all of these networks with pieces of string, you will notice that A,B,C and E all require the strings to be tied together in the same places, whilst D does not. Actually A,B,C and E are equivalent, so D is the odd one out.

2 marks

2.2 All maps can be turned into networks, but not all networks can be turned into maps. A network cannot be turned into a map if…

  • It has more lines than nodes.
  • It has an odd number of lines connected to any node.
  • The lines cross over each other, and cannot be untangled.
  • The lines cross over each other, but can be untangled.
  • There is a prime number of nodes.

3. Four-colour map theorem (part 3)

In this section, James dips into how they proved the four-colour map theorem. It is only a sketch outline of the proof, but you might still struggle to follow every detail – don’t worry, I am showing you this video just to give you a glimpse of one of the most interesting problems in maths. And I think you will have no problem answering the questions afterwards.

2 marks

3.1 The proof of the four-colour map theorem was a historic moment for mathematics, because it was the first mathematical proof that:

  • required a major input from a computer.
  • had taken over a century to find.
  • was discovered by two mathematicians working together.
  • won a Nobel Prize.
2 marks

3.2 The mathematicians found a proof by showing that the infinite number of possible networks (or maps) could be built from a finite number of nets (or maps). They then had to check the colourability of this finite number of networks. How many maps/networks did they have to check?

  • Roughly 10
  • Roughly 100
  • Roughly 1,000
  • Roughly 1,000,000
  • Roughly 1,000,000,000

4. Intermediate Maths Challenge Problem (UKMT)

3 marks

4.1. A standard pack of pumpkin seeds contains 40 seeds. A special pack contains 25% more seeds. Rachel bought a special pack and 70% of the seeds germinated. How many pumpkin plants did Rachel have?

  • 20
  • 25
  • 28
  • 35
  • 50

As a special pack contains 25% more seeds than a standard pack of 40, it contains 125100×40=54×40=50 seeds.

As 70% of these germinated, 70100×50=35 seeds germinated.

So Rachel had 35 pumpkin plants.

5. Intermediate Maths Challenge Problem (UKMT)

5 marks

5.1. Which diagram shows the graph y=x after it has been rotated 90° clockwise about the point (1, 1)?

  • A
  • B
  • C
  • D
  • E

The line y=x goes through the point (1, 1). So, if it is rotated about this point, the resulting line also goes through (1, 1). This means it could only be the graph in diagrams A or B. However, diagram B shows the graph of y=x. So, assuming that one of the diagrams is correct, it must be diagram A.

This answer is good enough for the IMC. However, to show that A really is correct, we need an additional argument. It is easily seen from the geometry that if we rotate the line y=x through 90° clockwise about the point (1, 1), we obtain the line shown in diagram A.

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.