[spoiler] With six races we can determine the fastest horse (lets call it A) but we cannot guarantee finding the second fastest (lets call it B). Suppose that we succeed in finding A. 24 horses must lose a race (that is finish second or lower) so that we know they aren’t the fastest. With six races there are only 24 opportunities to do this, so each horse loses precisely one race. This is the pigeon-hole principle. It is possible that the first race includes A but does not include B, and that some other horse, say C, finishes second. C has lost to A, so we know it isn’t the fastest, but it does not lose any other race so we will never find out that it is slower than B. We cannot rule out C as the second fastest horse and so we cannot identify that B is the second fastest. [/spoiler]

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And I thought it was 9. I thought I had to have all the second place horses run against each other, and all the 3rds. Then I ended up with the same last race. Now your solutions (BP and RMJ) make total sense to me.

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You only need 7 races.

First, race 5 heats of 5 horses each.

Then race the 5 winners against each other.

Let’s call the 5 heat winners A1, B1, C1, D1, E1 in the order that they finished the 6th race.

Then, A1 is necessarily the fastest horse, since every other horse has either lost to her or to a horse who’s lost to her. So we just need to determine the 2nd and 3rd place horses.

The contenders are:

A2, A3 (the 2nd and 3rd place horses in A1’s original heat)

B1, B2

C1

A2 and B1 are the only possible horses for second place, and the other one of these two plus the other three listed above could be the third fastest.

So, race these 5 in the 7th race to determine who is 2nd and 3rd.

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Yes please post your answers here. However use the spoiler tag if you think you have the actual answer. No need if it’s just thinking it through.

Spoiler tags are *[ spoiler ] my brilliant answer [ /spoiler ]*

Remove the spaces in the tags, I had to include them or the system would have treated it as a proper use of spoiler tags.

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