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# Find The Three Fastest

## Question

At a track meet, you have 25 runners for a 100 meter event and only 5 lanes. You've forgotten all of your timing devices at home. Assume that all runners {R1, R2,...R25} run the event in different times from each other, but an individual runner RN, will run the event in the same time each time he/she runs.

Your job is to figure out who is the 1st, 2nd and 3rd fastest runners. What is the minimum number of heats needed to make your determination?

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Run everyone in 5 heats to eliminate 4th and 5th place finishers in each heat. (10 runners)

In heat 6 run the five winners. This identifies the fastest runner overall. (1 runner) It eliminates 4th and 5th place finishers and all who placed behind them in previous heats (6 runners); anyone who placed behind the 3rd place finisher (2 runners); and the runner who placed 3rd in the heat won by the third place finisher. (1 runner). Six heats thus identify the fastest runner overall and eliminate 19 other runners.

Heat 7 among the five remaining runners identifies the 2nd and 3rd fastest overall.

One possible set of outcomes  (showing order of finish; x means eliminated)

1. a b c x x
2. d e f x x
3. g h i x x
4. j k l x x
5. m n o x x

6. a d g j m

Fastest overall: a
Eliminated: h i j k l m n o (all are slower than a d g) and f (slower than a d e.)
Remaining:
b c d e g.

7. b c d e g

Three fastest, in order: a b c.

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the answer is same like horse racing puzzle.

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