Problem 30 · 2016 Math Kangaroo
Stretch
Counting & Probability
casework
Exactly 2016 people are taking part in a conference. They are registered as P1 to P2016 in the system. Each person from P1 to P2015 has shaken exactly the amount of other hands that his/her own system number indicates. How many people did P2016 shake hands with?
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Answer: D — 1008
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Hint 1 of 2
P2015 shook everyone, so each person shares a handshake with the busiest people first.
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Hint 2 of 2
Pair the busiest with the least busy and peel inward; track who is left for P2016.
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Approach: pair from the extremes
- P2015 shakes everyone (so P1's single handshake is with P2015); P2014 shakes all but P1 (matching P2's two), and so on.
- Peeling these matched pairs inward, P2016 ends up shaking exactly half of the others.
- That is 2016 ÷ 2 = 1008.
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