The twenty five horses

1. Optimal strategy: Let's call the groups: A, B, C, D.
2. Candidates for positions 2 and 3: The first two are positions 2 and 3 overall.
3. Final section: Optimal strategy: Sorted in each group: A1 < A2 < A3 < A4 < A5 (and analogous for B, C, D, E). Then, analysis: A2, A3: they could be in the top 3.

Answer: 7 races.
Optimal strategy:
Races 1-5: Divide the 25 horses into 5 groups of 5. Race each group.

• Let's call the groups: A, B, C, D, E
• Sorted in each group: A1 < A2 < A3 < A4 < A5 (and analogous for B, C, D, E)

Race 6: The winners of each group run: A1, B1, C1, D1, E1

• Suppose result: A1 < B1 < C1 < D1 < E1

Analysis:

• A1 is the fastest horse (guaranteed top 1)
• D1 and E1 (and everyone from D, E) cannot be in the top 3
• C1: could be top 3, but C2, C3, C4, C5 cannot
• B1: definitely top 3, B2 could be top 3
• A2, A3: they could be in the top 3

Candidates for positions 2 and 3:

• A2, A3, B1, B2, C1

Race 7: Race these 5 candidates

• The first two are global positions 2 and 3

Result:

• Fastest: A1
• Top 3: A1 + the first two of race 7

Total: 7 races