25 Horses, 5 Lanes Puzzle

25 horses, 5 lanes per race, no timer — only relative order within a race is observable. What is the minimum number of races needed to identify the top 3 fastest horses? Requirements 25 horses; each race seats exactly 5; outcomes give only the in-race relative order, no times. Goal: identify the top 3 fastest overall with the fewest races. Classic interview puzzle — the answer is 7 races. Notes The standard derivation: Races 1-5: Race each disjoint group of 5 horses. Total 5 races. Each horse now has a within-group rank A1 > A2 > A3 > A4 > A5, B1 > B2 > …, etc. Race 6: Race the 5 group winners (A1, B1, C1, D1, E1). Without loss of generality, A1 > B1 > C1 > D1 > E1. The overall winner is A1. Eliminations after race 6: All horses in groups D and E cannot be in the top 3 (their group winners are 4th / 5th in race 6, so every horse below them is at worst 5th). In group C, only C1 can still be top 3 (since A1, B1 are faster). In group B, only B1 and B2 can still be top 3. In group A, A1 is locked in 1st; A2 and A3 can still be top 3. Race 7: Race A2, A3, B1, B2, C1. The top 2 finishers are the overall 2nd and 3rd. 7 is provably optimal: 6 races give only 30 pieces of ranking info, which is not enough to distinguish all 3-permutations of 25 horses by an information-theoretic argument. Variants the interviewer may layer on: "top 1 only" (answer: 6 races), "top 5" (answer: harder; typically 9-10 with case analysis). Preparation Be ready to walk the elimination logic out loud, layer by layer — the interviewer scores reasoning clarity more than the final number. Practice the "top 1" and "top 5" variants for follow-ups.