Problem 12 · 2011 Math Kangaroo
Stretch
Logic & Word Problems
careful-counting
Three racers take part in a Formula-1 race: Michael, Fernando and Sebastian. From the start Michael is in the lead, in front of Fernando, who is in front of Sebastian. During the race Michael and Fernando overtake each other 9 times, Fernando and Sebastian 10 times, and Michael and Sebastian 11 times. In which order do the three finish the race?
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Answer: B — Fernando, Sebastian, Michael
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Hint 1 of 2
Each overtake of a pair flips that pair's relative order, so an odd count flips it and an even count restores it.
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Hint 2 of 2
Apply the odd/even rule to each of the three pairs to get the final order.
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Approach: use parity of each pair's overtakes
- Start order M, F, S. Michael–Fernando swap 9 times (odd) → F ahead of M.
- Fernando–Sebastian swap 10 times (even) → F stays ahead of S.
- Michael–Sebastian swap 11 times (odd) → S ahead of M.
- So F first, then S, then M: Fernando, Sebastian, Michael, choice (B).
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