Problem 17 · 2023 Math Kangaroo
Hard
Logic & Word Problems
careful-countingcasework
A staircase has 2023 steps. Every third step is coloured black. The first seven steps of this staircase can be fully seen in the diagram. Anita walks up the staircase and steps on each step exactly once. She can start with either the right or the left foot and then alternates her right and left foot. What is the minimum number of black steps she sets her right foot on?
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Answer: D — 337
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Hint 1 of 2
Number the steps; the black ones are the multiples of 3: 3, 6, 9, …, 2022.
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Hint 2 of 2
Alternating feet means a step's foot depends only on whether its number is odd or even — so the two feet take the odd-numbered and the even-numbered black steps.
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Approach: split the black steps by parity
- The black steps are 3, 6, 9, …, 2022, that is 3·1 up to 3·674, so there are 674 of them.
- Feet alternate, so odd-numbered steps all use one foot and even-numbered steps the other.
- Among the black steps, the odd ones are 3·1, 3·3, …, 3·673 (337 of them) and the even ones are 3·2, 3·4, …, 3·674 (also 337).
- Either starting foot makes the right foot land on one of these groups, so the minimum is 337.
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