Problem 10 · 2020 Math Kangaroo
Medium
Counting & Probability
careful-countingcasework
Cynthia paints each region of the figure in a single colour: red, blue or yellow. Regions that touch each other must be painted different colours. In how many different ways can Cynthia paint the figure?
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Answer: E — 6
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Hint 1 of 3
Find which regions actually share a border, since only touching regions are forced to be different colours.
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Hint 2 of 3
Colour one region first, then count how many free colour choices are left for each region next to it.
Still stuck? Show hint 3 →
Hint 3 of 3
Multiply the number of choices region by region as you work inward.
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Approach: colour the regions one at a time and multiply the free choices
- Colour the outermost region first: there are 3 colours to pick from.
- Each region just inside it touches the one outside, so it must use a different colour, which usually leaves only a small number of choices.
- Multiplying the choices region by region as you move inward gives 6 different ways to paint the whole figure, choice E.
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