Problem 8 · 2020 Math Kangaroo
Medium
Counting & Probability
Geometry & Measurement
careful-countingcasework
Cynthia paints each region of the figure a single colour: red, blue or yellow. Regions that touch each other must be painted different colours. In how many different ways can she colour the figure?
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Answer: E — 6
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Hint 1 of 2
The picture is a chain of nested regions; neighbours sharing a border must differ in colour.
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Hint 2 of 2
Work along the chain: each new region just needs to avoid the colour of the one it touches, so multiply the choices.
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Approach: count colourings region by region with the touching rule
- The figure splits into regions; touching regions must get different colours from {red, blue, yellow}.
- Colour the regions one at a time: the first is free, and each following region only has to differ from the single region it borders.
- Multiplying the available choices at each step gives 6 valid colourings.
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