Parallel lines, quadrilaterals and triangles · Usually met around P3
Colouring a map
A ring: even needs 2 colours, odd needs 3.
How to do it
Method
A ring: even needs 2 colours, odd needs 3
A ring of 6: red, blue, red, blue, red, blue — 2
Another way
Try region by region
Reuse a colour wherever you can
In full
When colouring a map, regions that share a stretch of border must have different colours; touching at just one point doesn't count. First find regions that all touch one another — they need different colours — then go region by region, reusing a colour wherever you can. Regions in a row need only 2 colours, taking turns. Regions in a ring: an even number can take turns with 2; an odd number clashes on the way round and needs 3. A region in the middle touching the whole ring needs 1 more. Mathematicians have proved that 4 colours are always enough for any flat map.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
12 houses stand in a row, and neighbouring houses must be painted different colours. What is the fewest colours needed?
Show the answer and working
Answer: 2 colours
In a row, two colours taking turns are enough: red, blue, red, blue… for all 12.
Example 2
The regions of this map make a ring round a lake, which isn't coloured. Neighbouring regions need different colours. What is the fewest colours needed?
Show the answer and working
Answer: 2 colours
The ring has 4 regions: an even number takes turns with 2; an odd number clashes on the way round and needs 3.
Example 3
The outer regions of this map make a ring, and the middle region touches every one of them. Neighbouring regions need different colours. What is the fewest colours needed?
Show the answer and working
Answer: 4 colours
The ring of 3 needs 4 − 1 colours, and the middle touches them all, so 1 more: 4.
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