Quote:
Originally Posted by Hamlet53
So by request, maths . . .
You have 25 horses and want to know which are the three fastest. Whenever you race horses, the order of finish accurately reflects the relative speeds of the horses but you can only race five at a time. What’s the minimum number of races required to determine the three fastest, and how do you do it?
To be clear you have no measure of absolute speed, that is no race times. To forestall any similar questions you are sitting in a room without a view of the track and are only handed slips of paper with the names of the horses and the relative positions of finish after each race. You may select any five horses to participate in each race.
Provide your reasoning with the answer for any credit.
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First, split the horses into five groups of five, call them A,B.C.D & E groups. Race each group. Call the fastest horse in each group A1, B1, C1, D1, E1; the second fastest horse A2, etc.
Take the top three horses from each group - 15 horses in all. The fastest three horses of the 25 must be in this fifteen.
Split the fifteen horses into three groups:
Alpha Group: A1,B1,C1,D1,E1;
Beta Group: A2, B2, C2, D2, E2;
Gamma Group: A3, B3, C3, D3, E3
Race each group.
The fastest horse of all is the winner of the Alpha group.
The second fastest horse is either the horse who came second in the alpha group, or the horse that came first in the beta group
The third fastest horse is either the horse that came second or third in the alpha group, or the horse that came first or second in the beta group, or the winner of the gamma group.
That is, the second and third fastest horses are two out of five horses. Run those five horses in a race, and the first and second of that race are the second and third fastest overall.
Total number of races required: 5+3+1 = 8.
(The trick is to realise that after the second lot of races, we can already see who the fastest horse is.)