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Puzzles for Spies – from GCHQ

We looked at a lovely way to encode secret messages from a fantastic book published by GCHQ.

Here are some instructions to code using this method on excel

Challenge

A country was used as the key instead of DOLPHIN.  Given the following encodings, can you guess which country?

CHINA- 0FJBD

SCOTLAND – T0WSYDBA

KENYA – Z9BLD

BRUNEI – NGQB9J

How many ways can circles overlap

This activity was inspired by this numberphile video by Neil Sloane.

Here are the only distinct three ways you can draw two circles in a Venn diagram.

If we take the universal set to be the positive integers, we can come up with some sets that would fit for A and B for the three examples.

  1. A = Odd numbers, B = Multiples of 4
  2. A = Even numbers C= Prime numbers
  3. A= Even numbers B=Multiples of 4

Note that these choices mean that there is at least one number in each region.

Now for the challenge …

Draw all the different Venn diagrams you can make with three circles, and find rules for them all.

Number of ways can be found here. A picture of them all is here.

Watch the numberphile video to find out how many Jonathan Wild found for 4 and 5 circles. It is a lot more! Nobody knows for 6 …

Binary division rules

You might be familiar with divisibility rules such as “A number is divisible by 3 if the sum of the digits is divisible by 3”. You can see some more here.

Our investigation today was – can we develop a set of divisibility rules for binary numbers?

Divisible by 2

This was straightforward. If the rightmost (unit) digit is 0 then the number is divisible by 2.

e.g. The number 6 is divisible by 2

but the number 7 is not

Can you extend this logic to create a divisibility rule for powers of 2 (e.g. 4, 8, 16 …)? Solution

Hadwiger-Nelson problem

This activity came from watching this Numberphile video . A worksheet of the activities below is available here.

Suppose that we want to colour the whole plane so that any two points at distance 1 from each other would have different colors.

You can imagine this as walking on a tiled floor in jumps of 1 and each jump has to be onto a different colour.

What is the minimum number of colours we would need to use?

Finding an upper bound to the number of colours needed

Task 1

It doesn’t have to be a regular tiling, but would 4 colours work if we coloured the plane like above? How big would the squares have to be?

Task 2

Find a number of colours that would work for a square tiling. How big would the squares have to be?

Task 3

Show that we would need at most 7 colours using a different shape tile (isometric paper will be helpful!).

Finding an lower bound to the number of colours needed

Task 4

Use this diagram to show that we would need at least 3 colours to colour the plane.

Task 5

Create a graph of points to show that we need at least 4 colours.

Further reading 

This problem is called the Hadwiger–Nelson problem and was stated in 1950. It was recently (2018!) proved that you need at least 5 colours by an amaetur mathematician who was working on it as a break from his job as a maverick biologist intent on extending the human lifespan! The middle dot in the picture below has to be white rather than red, green, blue or yellow like the rest.  It is still an open problem to find the smallest number of colors. It could now be 5, 6 or 7 and nobody knows.

Links:

https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/

https://www.theguardian.com/science/2018/may/04/60-year-old-maths-problem-partly-solved-by-amateur

https://mathworld.wolfram.com/Hadwiger-NelsonProblem.html

The chromatic number of the plane, part 4: an upper bound