Problem 22 · 2024 Math Kangaroo
Stretch
Number Theory
careful-countingoff-by-one
Daniel wants to cut a rope into 12 equal pieces and marks the places where he has to cut. Mohammed wants to cut the same rope into 16 equal pieces and marks his cuts as well. Maya finally cuts the rope at all the marked spots. How many pieces does Maya get?
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Answer: A — 24
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Hint 1 of 2
Count the marks each boy makes (11 for twelve pieces, 15 for sixteen pieces), but watch for marks that land on the same spot.
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Hint 2 of 2
Shared marks happen where the fractions agree; subtract those once, and remember the number of pieces is one more than the number of cuts.
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Approach: inclusion-exclusion on the cut marks, then add one
- Daniel makes 11 marks (at 1/12, 2/12, …, 11/12) and Mohammed makes 15 marks (at 1/16, …, 15/16).
- Marks coincide where the positions match, which happens at 1/4, 1/2 and 3/4 — that is 3 shared marks.
- Distinct cut points: 11 + 15 − 3 = 23.
- Cutting at 23 points gives 23 + 1 = 24 pieces.
Mark:
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