# How Many Races to Find the Top 3 Horses from 25 with Only 5 Tracks?

• MHB
• Albert1
In summary, you would need a total of 7 races to ensure that the fastest 3 horses can be chosen without using a stopwatch. This is because in each race, 5 horses can run at the same time, so it would take 5 races to narrow down the field to 5 horses. In the 6th race, the 5 winners from each previous race would compete against each other, leaving only 3 horses. And in the 7th race, the 3 remaining horses would compete against each other to determine the fastest 3. This method is known as the minimax strategy and guarantees that the 3 fastest horses will be chosen.
Albert1
you have 25 horses and you have to pick fastest 3 out of the 25. In each race
only 5 horses can run at the same time as there are only 5 tracks. what is the
minimax number of races to ensure the 3 horses can be chosen without using a stopwatch ?
(suppose the speeds of all horses are different)

Last edited:
Albert said:
you have 25 horses and you have to pick fastest 3 out of the 25. In each race
only 5 horses can run at the same time as there are only 5 tracks. what is the
minimax number of races to ensure the 3 horses can be chosen without using a stopwatch ?
(suppose the speeds of all horses are different)
my solution:
after each race two horses will be eliminated from competition ,
so when race 8 is finished ,there are only 9 horses remained marked with $A_1,A_2,A_3,A_4,A_5,A_6,A_7,A_8,A_9$
we arrange race 9:$A_1,A_2,A_3,A_4,A_5$
race 10: $A_1,A_2,A_3,A_6,A_7$ $(A_4,A_5)$ out from race 9
race 11:$A_1,A_2,A_3,A_8,A_9$ $(A_6,A_7)$ out from race 10
after race 11 the top 3 can be produced
$25-11\times 2=3$

Last edited:

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