25 Horses

There are 25 horses and you want to find the fastest 3 among them. You can race any 5 of the horses against each other and see the final standing, but not the running times. If all the horses have constant, permanent speeds, how many races do you need to organize in order to find the fastest 3?

Let us label the horses H1, H2, H3, H4, …, H24, H25.

We race H1 – H5 and (without loss of generality) find that H1 > H2 > H3 > H4 > H5. We conclude that H4, H5 are not among the fastest 3.

We race H6 – H10 and (without loss of generality) find that H6 > H7 > H8 > H9 > H10. We conclude that H9, H10 are not among the fastest 3.

We race H11 – H15 and (without loss of generality) find that H11 > H12 > H13 > H14 > H15. We conclude that H14, H15 are not among the fastest 3.

We race H16 – H20 and (without loss of generality) find that H16 > H17 > H18 > H19 > H20. We conclude that H19, H20 are not among the fastest 3.

We race H21 – H25 and (without loss of generality) find that H21 > H22 > H23 > H24 > H25. We conclude that H24, H25 are not among the fastest 3.

We race H1, H6, H11, H16, H21 and (without loss of generality) find that H1 > H6 > H11 > H16 > H21. We conclude that H16, H21 are not among the fastest 3.

Now we know that H1 is the fastest horse and only H2, H3, H6, H7, H11 could complete the fastest three. We race them against each other and find which are the fastest two among them. We complete the task with only 7 races in total.

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  1. With the proposed solution it is impossible to know if H2 is faster than H6 or not ( and if H7 is faster than H11; H12 faster than H16 and H17 faster than H21)

    1. Hi Almeida. In your last race, you set H2, H3, H6, H7, H11 against each other. After the first 6 races, you already know that H12, H16, H17, and H21 are not in TOP 3.

  2. Since we do not know the individual speeds of each horse, we cannot eliminate

    1. Thank you for your answer. It is true you don’t know the individual speeds of the horses, but you can find their relative speeds, assuming they are constant.