#### Question

You have 25 horses and you need to pick fastest 3 horses out of those. In each race you can race maximum of 5 horses as there are 5 tracks available. What are minimum number of races needed to find the fastest 3 without using a stopwatch.

#### Solution

**7 races needed.**

#### Understanding

Since we don't have a stopwatch only way to find fastest is by racing horses. Lets have 5 races each of 5 horses and let following be the results -

- A > B > C > D > E
- F > G > H > I > J
- K > L > M > N > O
- P > Q > R > S > T
- U > V > W > X > Y

**We have 5 races till now**. Now lest race between fastest of all previos 5 races i.e A,F,K,P,U.

**We have 6 races till now**.Let's say the result for that is -

- F > K > A > P > U

Since We are interested in top 3 P and U are useless for us. Also since P and U were fastest among their group we can ignore all members of P and U too. So now only horses under consideration are -

- F > G > H > I > J
- K > L > M > N > O
- A > B > C > D > E

Now horses under consideration are -

- F > G > H > I > J
- K > L > M > N > O
- A

- F > G > H > I > J
- K > L
- A

- F > G > H
- K > L
- A

These are 5 horses we can have another race and find the top 2 out of them. Lets say the result was -

- L > H > K > G > A

**We have done 7 races now**. So fastest onces are L and H. We already the fastest among all - F. So the final answer is -

- F > L > H