Problem 4 · 2024 Math Kangaroo
Medium
Counting & Probability
careful-countingoff-by-one
A type of hopscotch is played like this: a player jumps from one square to the next, alternating left foot, both feet, right foot, both feet, and so on, as shown. Maya plays and jumps into exactly 48 squares, starting with her left foot. How many times is her left foot on the floor during the game?
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Answer: C — 36
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Hint 1 of 2
The only jumps where the left foot is NOT on the floor are the right-foot-only jumps, so it may be easier to count those.
Still stuck? Show hint 2 →
Hint 2 of 2
Group the jumps into the repeating block left, both, right, both and see how the left foot behaves in each block.
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Approach: count the jumps where the left foot is down
- The pattern repeats every four squares: left, both, right, both.
- In each block of four, the left foot is down on the 'left' square and on the two 'both-feet' squares, so 3 of every 4 squares use the left foot.
- Over 48 squares that is \(\frac{3}{4}\times 48 = 36\) times, answer C.
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