The 25 Horses Puzzle: Find the Top 3 in 7 Races

⏱ 1 min read

You have 25 horses and you want to find the three fastest. You have a racetrack with 5 lanes, so you can race 5 horses at a time and see their finishing order. There is no stopwatch — you only learn the relative order of the horses in each race, not their times. Horses run consistently (a horse’s speed never changes from race to race).

What is the minimum number of races you need to guarantee you’ve found the top 3, in order?

This one shows up constantly at Google, and it’s a favorite at quant firms like Jane Street and Citadel. It’s a great puzzle because the “obvious” answer is wrong in a way that’s easy to talk yourself into, and the real answer requires you to notice which horses you can eliminate without racing them again. The no-stopwatch rule is the whole trick — if you had a timer, you’d just run 5 races, record times, and sort. Take a minute before revealing the strategy.

💡Strategies for Solving This Problem

Start by ranking within groups

Split the 25 horses into 5 groups of 5 and race each group once. That's 5 races, and now you know the internal ranking of every group — but nothing across groups yet.

Race the winners

The fastest horse overall must be the winner of its own group, so the 5 group-winners are the only candidates for 1st. Race those 5 against each other. After this race you know the single fastest horse for certain — and, more importantly, you can throw out a lot of horses.

The real insight: eliminate, don't re-race

A horse can only be in the top 3 if at most 2 horses are known to be faster than it. Use that rule ruthlessly on the results of the winners' race:

If a group's winner finished 4th or 5th among the winners, nobody in that group can be top 3 — drop all 10 of those horses.
In the remaining groups, count how many horses are already proven faster than each horse. Anything with 3+ faster is out.

You'll find only a handful of horses survive. One final race settles 2nd and 3rd. Now count your races.

✅Solution

The answer: 7 races

Races 1–5 — group stage. Split the horses into 5 groups (A, B, C, D, E) and race each group. Within each group you now have a full ranking; write A1 > A2 > A3 > A4 > A5 for "A1 is fastest in group A," and likewise for the others.

Race 6 — the winners' race. Race the five group winners: A1, B1, C1, D1, E1. Suppose the result is:

A1 > B1 > C1 > D1 > E1

Two things follow immediately:

A1 is the fastest horse overall (it won its group and beat every other group's best). That's your #1 — locked in.
Groups D and E are completely eliminated. D1 lost to A1, B1, and C1, so at least 3 horses are faster than D1 — D1 can't be top 3, and everyone in group D is slower than D1. Same logic for E. That's 10 horses gone without another race.

Who's left for #2 and #3?

Only a horse with 2 or fewer proven-faster horses can still place. Check the survivors:

A2, A3 — only A1 is proven faster, so each could be 2nd or 3rd.
B1 — only A1 beat it, so it could be 2nd.
B2 — slower than B1 and A1 (2 faster), so it could be 3rd.
C1 — slower than A1 and B1 (2 faster), so it could be 3rd.

Everyone else is out: A4/A5 have 3 faster (A1, A2, A3); B3 has 3 faster (A1, B1, B2); C2 has 3 faster (A1, B1, C1); and all of C3+, D, E are gone. That leaves exactly five candidates: A2, A3, B1, B2, C1.

Race 7 — the deciding race. Race those five. The top two finishers are your overall #2 and #3 (A1 is already #1 and isn't in this race). Done.

Total: 5 + 1 + 1 = 7 races.

Why 6 races isn't enough

After race 6 you know #1 for certain, but #2 and #3 are genuinely ambiguous: A2, B1, and the others have never been compared head-to-head. For example, A2 (second in the fastest group) and B1 (best of the second group) could be in either order — nothing so far decides it. You need the 7th race to break that tie, and one race over the 5 survivors is enough to settle both spots.

Why you can't do better than 7

You need at least 5 races just to run every horse once (25 horses, 5 per race). You need a 6th to compare across groups at all. And you need a 7th because a single cross-group race (race 6) can't rank the five remaining contenders for 2nd and 3rd. So 7 is optimal.

The variations interviewers tack on

"Just the fastest one?" 6 races: 5 group races plus one race of the winners.
"What if you had a stopwatch?" 5 races — record every time and sort. Pointing this out shows you understand that the no-timer rule is what creates the puzzle.
"Top 3 out of N horses, K lanes?" The pattern generalizes: race ⌈N/K⌉ groups, race the winners, then reason about which horses still have ≤2 faster. Walking through the general version is what separates a strong candidate from someone who memorized "7."

That last point matters in a real interview: the interviewer usually knows you might have seen this. The way to stand out is not blurting "7" — it's deriving it cleanly and handling the follow-up where they change the numbers.

Frequently Asked Questions

Why is the answer 7 and not 11?

11 comes from over-racing — re-running horses you've already eliminated. The key economy is that the winners' race (race 6) lets you discard 10 horses instantly and prove that only 5 horses can possibly be 2nd or 3rd, so a single final race finishes the job.

Do we assume horses run the same speed every race?

Yes — that consistency is a stated assumption, and it's what makes elimination valid. If horses had random off-days, no fixed number of races could guarantee the top 3.

Is this really asked at Google?

It's one of the most frequently reported Google and quant-finance puzzles. Because it's well known, interviewers care less about the number and more about whether you can derive it and adapt it when they change the horse and lane counts.