# Fastest 3 out of 25 Horses Problem

## Problem Statement

You have 25 horses, and you want to pick the fastest 3 horses out of those 25. Each race can have maximum of 5 horses at the same time. What is the minimum number of races required to find the 3 fastest horses without using a stopwatch?

## Solution

Let’s say that we have 5 races of 5 horses each, so each row in the table above represents a race.

 H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15 H16 H17 H18 H19 H20 H21 H22 H23 H24 H25

Let each row represent a race.

Step 1: Perform 5 races of each set.

Result:

 1st 2nd 3rd 4th 5th H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15 H16 H17 H18 H19 H20 H21 H22 H23 H24 H25

Step 2: Elimination by logical analysis:

• We can eliminate the slowest 2 horses in each group since those horses are definitely not in the top 3
• The 5 group leaders are not necessarily the 5 fastest horses, therefore race those 5 horses against each other (H1, H6, H11, H16, and H21) {Race 6}, Let’s say that the 3 fastest in that group are H1, H6, and H11 – automatically we can eliminate H16 and H21 since those 2 are definitely not in the top 3
• We can automatically eliminate all the horses that H16 and H21 competed against in the preliminary races as they were slower than H16 and H21
• We also know that H1 is the fastest horse in the group since he was the fastest horse out of the 5 group leaders
• if H6 and H11 are the 2nd and 3rd fastest in the group leaders, then we should be able to eliminate H8 since H6 raced against him and he was in 3rd place in that race
• We can also eliminate H12 and H13 since H11 was the 3rd fastest in the group leaders, and H12 and H13 were slower than H11
• This leaves us with the following horses to determine the 2nd and 3rd fastest horses:
 H2 H3 H6 H7 H11

Race the last Set {Race 7} to get the Top 2nd and 3rd racers with H1 as the fastest.

Total number of Races: 7

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