Problem 23 · 2019 Math Kangaroo
Stretch
Counting & Probability
careful-countingcasework
Natascha has some blue, red, yellow and green sticks, each 1 cm long. She wants to make a 3 × 3 grid, as shown, so that the four sides of every 1 × 1 square in the grid are all different colours. What is the smallest number of green sticks she can use?
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Answer: C — 5
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Hint 1 of 2
Each of the nine unit squares must use four different colours on its four sides.
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Hint 2 of 2
Push to use as few green sticks as possible while still colouring every square legally.
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Approach: minimise one colour under the per-square constraint
- Every 1×1 square needs its four sides in four different colours, so each square uses green on at most one side.
- Sticks are shared between neighbouring squares, so one green stick can serve two squares at once.
- Arranging the green sticks to cover all nine squares needs a minimum of 5 green sticks.
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