# Colouring curves game

This game is for two players, and you will need paper and a pencil.

Start by drawing a curve on the paper. The curve can cross over itself as many times as you like, but must join back to where you started (this is called a closed curve).

Here is an example of the sort of curve you might draw:

Image

Now take it in turns to
choose a region and colour it in. The only rule is that shaded
regions can't share an edge, although it's ok for them to meet at a
corner or vertex.

This is what the curve above might look like after each player has
had two turns:Image

Eventually, you will run
out of regions to shade without shading regions which share an
edge. The last person who can shade a region is the winner!

#### Here are some ideas to think about as you play:

- Is it better to go first or second? Does it depend on the curve?
- Can you design some simple curves where you can guarantee that you will win?
- What do you notice about the number of regions that meet at each vertex?
- Can you come up with any strategies to help you to win?

Another way to play the
game is to take it in turns to shade regions each using a different
colour, without shading adjacent regions in your own colour, so the
game might look a bit like this:

Image

### Why play this game?

This game provides an opportunity for learners to consider strategy and thinking ahead. Playing the game could provide a starting point for reading about mathematical ideas such as the Four Colour Theorem.### Possible approach

The game works well when
played in pairs. Learners could use pencil and paper, or perhaps
whiteboards. Once everyone has had the chance to play the game a
few times, the class could discuss any strategies they came up with
and explain anything they noticed while playing the game.

If learners play the two
colour version described at the end of the problem, they could
create some intriguing images for classroom display.

### Key questions

Is it better to go first or second? Does it depend on the curve?Can you design some
simple curves where you can guarantee that you will win?

What do you notice about
the number of regions that meet at each vertex?

Can you come up with any
strategies to help you to win?

### Possible extension

The game could be
investigated using various different representations for the curves
- one example is a graph such as this:

Image

By using the rule that no
two connected numbers can be coloured the same, learners could
experiment with different curves and investigate the idea that all
such curves can be coloured with just two colours.

Further investigation of
these ideas at a higher level can be found in the Stage 5 problem
Painting by
Numbers.