How can you sort out your top 3 fastest horses?
You’ve recently bought 25 race horses from where you are to choose the fastest 3 for the upcoming competition.
Nearby your stable of horses there is a racing track with five lanes only, i.e., only five horses can run in the track at a time and each time you have to pay a good amount of money to the owner of the track for using it.
You only know that different horses has different running speed and the running speed of each horse remain constant whenever they run in a race. But you don’t have any idea about the individual running speed of any horse among the 25.
Now you can find out the fastest 3 horses by using the race track. You can use the track as many times as you want, but you have the constraint of time and money. Therefore, you have to use the track the minimum possible of times so that you will have to pay the minimum possible amount of money for the use of the track and will spend the minimum possible amount of time.
After each race the track owner provide you only the rank of each horse in the race, so you don't know how much time each horse took to complete the race.
Now the question is, what is the minimum number of times you have to use the race track so that you can find out the fastest 3 horses?
The answer is 7.
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1 solution
Let first send the horses to the track in groups of 5 and name each group A, B, C, D, and E and name each horses according to their ranking in the group. For example, C3 imply the horse which is ranked 3 in the group C. So up to now there are 5 races.
Now let's have the 6th race among the fastest horses in each group (A1, B1, C1, D1, and E1). Let assume that the top three horses in the race are A1, B1, and C1 (you can choose any other combination an proceed accordingly). Since A1, B1, and C1 are the top three horses in the 6th races, we can easily eliminate all the horses from group D and E.
Now we can eliminate the horses A4, A5, B4, B5, C4, and C5 as they cannot be among the top 3.
From the 6th race we have already certain that A1 is the fastest horse among the 25, but we are not sure about 2nd and 3rd.
Now we have to find out which horses among the remaining can not be at least 3rd.
Here B3, C2, and C3 can never be at least 3rd
Finally, the 7th race will be among A2, A3, B1, B2, and C1. Along with A1, top 2 in the 7th race will give us the top 3 horses among the 25