This is the final maths blog of the academic year. So to mark this special occasion I have asked Mrs Hannah Hukins to be a guest blogger. For those who do not know her, she is both a a parent and teaching assistant. In addition, she has a maths degree and simply LOVES maths!

### Mrs Hukins: This week’s challenge involves colour!

This puzzle is all about colouring in using the fewest colours possible without any sections next to each other being the same colour. Take a look at this pattern.

We could colour it in like this using 4 colours

But could we use fewer than 4 colours and still not have any sections touching with the same colour? How about this?

I don’t think we can use less than 2 colours can we?

Now have a go with the following patterns – do as many or as few as you like. But remember, no sections next to each other can be the same colour, but you’re trying to use as few colours as possible.

Maybe you could try drawing your own patterns and colouring them in.

This bit of maths is useful for people who make maps – take a look at this map of the world, how many different colours have been used?

It’s not many is it, given that there are nearly 200 countries in the world!

Send your answers to [email protected]

## I look forward to seeing your lovely colouring in!

Mrs Hukins

#### NEXT ONE – KS2 Come along…..

Quirkville is a town which lies on the river Grimble, it has 2 islands, connected by 5 bridges as shown below.

The people of Quirkville like walking but they are wondering is it possible to walk along ALL the paths and bridges and get back to their starting point without walking along the same path or bridge twice? We could draw a simple diagram to represent the islands, paths and bridges as shown here

To answer the people of Quirkville’s question, we could see whether we can draw the shape that diagram makes without taking our pencil off the paper…. have a go and see whether you can? Try starting at different points? Can you do it if you start at point A? What about point F?

Mrs Gilhouley and Mrs Mackelworth both live in Quirkville, on the river banks; they decide to build bridges from their houses to their favourite island…

Now can the people of Quirkville walk along every path and bridge without walking the same path or bridge twice? Again we can draw a simple diagram and see if we can draw the shape it makes without taking our pencil off the paper, is it possible?

Did you manage it?

Without the extra bridges it IS possible if you start at point F or C

This is one way of doing it

But with the extra bridges added it is NOT possible…. Why is that? Let’s look at some other shapes and see if you can draw them without taking your pen off the paper, how about this one – it is possible, challenge your friends to see whether they can do it!

Can you do it? Hint – try starting at point D or E

To work out why it works with some shapes and not others, we need to look at the points where the lines (or paths and bridges) meet – the mathematical word for the points is the vertices (the singular of vertices is vertex) – and we need to see how many lines come out of each vertex. If we look back to our first diagram there are 6 vertices (A – F); 4 vertices have 2 lines coming out of them (A, B, E and D) and 2 vertices have 3 lines coming out of them (F and C). If you remember, we had to start drawing from F or C, why is that?

Look at these shapes and then fill in the table below.

Shape number | How many vertices? | How many vertices with an odd number of lines? | How many vertices with an even number of lines? | Can you draw it without taking your pencil off the paper? |

1 | ||||

2 | ||||

3 | ||||

4 | ||||

5 | ||||

6 | ||||

7 | ||||

8 |

The big question is, can you work out how you can tell whether you will be able to draw the shape without taking your pencil off the paper or not? Try drawing some more complex shapes that will work. Send your answers to [email protected] you can also email me if you have any questions or want a hint.

I hope you enjoy this puzzle – this is part of an area of maths called graph theory which I actually studied at university.

Mrs Hukins