Can you solve the fastest horse 🐴 algorithm problem?

Start by grouping them into 5 groups - a, b, c, d and e - and have each group race within itself. Let's say these are the results:

a1 b1 c1 d1 e1
a2 b2 c2 d2 e2
a3 b3 c3 d3 e3
a4 b4 c4 d4 e4
a5 b5 c5 d5 e5

Obviously any horse ranked fourth or fith within its own group cannot be in the top 3, leaving us with:

a1 b1 c1 d1 e1
a2 b2 c2 d2 e2
a3 b3 c3 d3 e3

Now, pick the fastest horse of each group and let them race among each other. Without loss of generality, assume a1 finished first, then b1, then c1, then d1 and finally e1.

d1 and e1 cannot be in the top 3, and we can also eliminate the rest of the d and e groups (which are slower than d1 and e1):

a1 b1 c1
a2 b2 c2
a3 b3 c3

c1 cannot be the fastest horse, nor can it be the second fastest - at most it can be third, because a1 and b1 are faster than it. This means c2 and c3 can at most be fourth and fith - which we do not care about:

a1 b1 c1
a2 b2
a3 b3

Similarly, we can eliminate b3 - but not b2 because it can still be third (if the first trio is a1 b1 b2):

a1 b1 c1
a2 b2
a3

This leaves us with 6 horses. But we already know that a1 is the fastest of them all - so we can just make the other five race to determine second and third place.

A total of 7 races.

Given there is no constraints on what information we can gather from each race, and no constraints on assumptions. The only constraint is the number of tracks we have to use.

We can do this in five races providing we can time the races and each horse runs it's fastest every time it runs.

Split the horses into 5 groups and ensure that every horse runs at least once.

Time the horses and the 3 fastest times give you the 3 fastest horses.

7 races? That doesn't seem right.
Split the 25 horses into 5 arbitrary groups of 5. For each group, determine the fastest horse. Race each group. You now have 5 horses, each the fastest of their group. Race those 5 horses to determine the fastest horse overall - a total of 6 races.

EDIT: I misread the question statement, so the above answer is incorrect. Sorry about that. I would still encourage you to solve my question down below (it's easy and fun!), though it gets a little trickier when you want to know the j fastest horses instead of just the fastest :)

So each horse runs the given race in the exact same amount of time every time (individually, not collectively)? There exists absolutely zero overlap of +/- of average time between horses such that given the same two horses, horse one could beat horse two or just as easily vice versus? That’s the only way you can say the generalized solution holds. And once you realize that you understand how unrealistic the problem in general is. It’s just academic nonsense.

I remember going over this problem in class, it was a good time and we eventually came up with the same solution as stated by @idanarye . There was caveat that was brought up that should be mentioned and that's we shouldn't have a way to track individual race horse times.

Otherwise I could solve this problem by stating "time every horse using a stop watch", thus 5 races are the minimum to "get the speed" of every horse.

The stop-watch approach is 0 fun, but it is worth pointing out a missing constraint could totally change the problem.

I love puzzles/problems like this as they wreck your brain, and yet make your really think 😄

Here's the solution I thought of. Assuming we're not allowed to measure the speed and judge whether a horse is fast or not depending on whether it finishes first or not.
Conduct five races with groups of 5 horses and eliminate the last two finishers from each group, which will leave us with 15 horses. Repeat the process again - but this time it takes three races - and we are left with 9 horses. Repeat the process again - eliminate last 2 horses from the group of 5 and last 1 horse from thr group of 4 - and we're left with 6 horses, 2 races taken. And finally, take 5 from these, get the top 3 (1 race taken) and race it with the other one left and eliminate the last one leaving with 3 horses(another race taken).

Means a total of 12 races. Now lemme check the video...
Happy Coding!