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Colouring Curves Game

Stage: 2 and 3 Challenge Level: Challenge Level:1

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:

closed curve
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:
shaded regions
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:

coloured curve