Assume you have 8 horses and you can race any 5 of them at time. After every race you get the 3rd and 4th fastest among them. How will you find the fastest among them? What would be the general approach if this was n horses?
More than the solution I wanted to know your intuition or approach for this type of problems.
The task is impossible.
One approach:
As you've set the problem, it's not possible to identify the fastest horse. It will always be impossible to distinguish between the fastest and the second-fastest horse, regardless of what races you run.
In any race with both, they will come first and second respectively. As you are only told which horses come third and fourth, you will never see these horses in the race results. Whether they come first-and-second or second-and-first, the race results are identical; since they can be swapped without changing visible results, they are indistinguishable.
In any race with only one, they will come first. As you are only told which horses come third and fourth, you will never see these horses in the race results. Again, whether they come first-and-not-entered or not-entered-and-first, the race results are identical; since they can be swapped without changing visible results, they are indistinguishable.
In your limited results view, for every single race, it is impossible to distinguish between the two of them.
Another approach:
Previously, I mentioned that because the slowest horse will never be in the results, it is also indistinguishable. This is incorrect, as Deusovi proves in the linked duplicate.
While it is true the slowest horse will never be in the results, it is possible to use the information to still identify it. For more details, see Deusovi's answer here.
After each race we get two horses that for sure aren't the fastest. So we can take them out and race another set of 5 and so forth. However, I don't know how we can make a difference between the top 2 horses (who are never 3rd/4th) and the slowest one (who's also never 3rd/4th).
