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# hidden race order

## Question

The 400 metre dash sprinting event will be held at a field track with 5 lanes. 25 athletes will be participating in total, of which, obviously, only 5 can be running together at a time. Define the minimum number of dashes required to determine the 3 fastest athlets of all, so that they are awarded the gold, silver and bronze medal. Which athletes will be running in each dash?

We assume that each athlete performs exactly the same in each dash. The results of the event will be determined by the relative classification of the athletes and not by their exact times. We only need to determine the 3 fastest athletes and not to follow the exact procedure which usually is followed at such events. (Obviously we cannot use a stopwatch).

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## 4 answers to this question

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The fastest 5 runners can be determined in 7 heats.
It is clear that 5 heats are needed for all to run at least once.

So the question seems to be can we determine the top 3 in just 6 heats.

The answer seems to be no.

The three fastest wight all be in one of the first 5 heats.

So the logical 6th heat among winners of the first 5 heats won't suffice.

I'll wait for discussion before posting the 7-heat solution.

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The fastest 5 runners can be determined in 7 heats.

....

I'll wait for discussion before posting the 7-heat solution

Please post, I think that you cannot get more than 3:

Make them run 5 by 5, attribute a letter to each race and number them in the order of arrival. You get:

A1 A2 A3 A4 A5

B1 B2 B3 B4 B5

...

E1 E2 E3 E4 E5

6th heat: Make run A1, B1, C1, D1, E1. If they do not arrive in that order, re-letter them so A1 is faster then B1 who is faster than C1...

Gold medal: A1

Other medals: make run (7th heat): A2, A3, B1, B2, C1.

If they arrive in this order, the 4th fastest can be A4, D1...

Edited by harey

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sounds like the classic 25 horse problem

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My memory was faulty. Only the three fastest can be determined in seven heats.

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