You have 25 uniformly capable horses and a track that can race exactly 5 horses simultaneously. You do not have a stopwatch, meaning you can only rank horses relative to their specific race. What is the absolute minimum number of races required to identify the fastest 3 horses out of the 25?
Split the 25 horses into 5 groups of 5:
Group A
Group B
Group C
Group D
Group E
Race each group once.
After 5 races, suppose the rankings are:
A1 > A2 > A3 > A4 > A5
B1 > B2 > ...
C1 > ...
D1 > ...
E1 > ...
(Where A1 means winner of Group A, etc.)
So far: 5 races
Now race:
A1, B1, C1, D1, E1
Suppose the result is:
A1 > B1 > C1 > D1 > E1
So A1 is the fastest horse overall.
Total so far: 6 races
Now determine who can still possibly be in the top 3 overall.
Since:
D1 lost to A1, B1, C1
E1 lost to A1, B1, C1, D1
No horse from D or E can be top 3.
Eliminate all horses in D and E.
C1 lost to A1 and B1.
So only C1 can still possibly be 3rd.
C2 and below cannot.
Keep only:
C1
B1 could be 2nd.
B2 could still be 3rd.
But B3 and below cannot.
Keep:
B1, B2
A1 is already confirmed fastest.
A2 could be 2nd.
A3 could be 3rd.
Keep:
A2, A3
We already know:
1st place = A1
Candidates for 2nd and 3rd:
A2
A3
B1
B2
C1
Exactly 5 horses remain uncertain.
Race:
A2, A3, B1, B2, C1
The top 2 finishers in this race are the overall 2nd and 3rd fastest horses.
This is the 7th race.
