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Two coloured drawings

By , 12 March 2014

How many coloured pencils or textas do you really need to own? Can you get away with using just two? Time to get colouring with this maths activity.

someone drawing a blobby loop shape.

If your line starts inside the frame, end it at the same point so it makes a closed loop.

You will need

  • Paper
  • Pencil or pen
  • Two coloured pencils or textas

What to do

  1. Take your pencil and paper and draw a frame for your picture.
  2. You’re going to draw some lines in the frame, but you need to follow some rules:
  • If you start your line on the frame, you need to end it on the frame.

    Someone is drawing a straight line across the page.

    If you start on the frame, you can end your line anywhere else on the frame.

  • If you start your line somewhere inside the frame, you need to end it where you started.
  • When you meet another line, you have to cross it – you can’t trace along the old line.
  1. Draw as many lines as you like, following the rules above.
  2. When you’ve finished drawing lines, look at your drawing. The space inside the frame will be divided up into regions.
  3. Now colour the regions, but if two regions are next to each other (they share an edge), you have to colour them different colours. Two regions of the same colour are allowed to meet at an intersection. Can you colour them all with only two colours?

    Someone is colouring shapes in yellow and blue.

    If to areas share an edge, colour them different colours.

What’s happening?

If you follow the rules and you’re careful when you colour, you should be able to finish with only two colours. Your picture is ‘two colourable’!

Not all pictures are two colourable. Imagine a circle cut into three segments, like a pizza. Each piece is next to the other two. If you colour one black and a second white, there’s no colour to paint the third. You’ll run into similar problems where any odd number of regions meet at an intersection.

a picture coloured in with blue and yellow.

You’ll end up with a work of art!

If you follow the rules in this activity properly, there should be an even number of regions meeting at every intersection inside the frame. Intersections on the edge of the frame might have an odd number of regions, but they don’t matter – both colours can border on the outside region.

This explanation isn’t quite complete. Try drawing a figure-8. How many regions meet at the middle intersection? Can you colour it while still following the rules?

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