Horse Race
Logic
Level
3
There are 25 horses in my garden and I want to figure out the fastest, second fastest and the third fastest horse by conducting races.
If a maximum of 5 horses can participate in a race, what is the minimum number of races I need to conduct?
8
9
6
12
11
5
7
10
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1 solution
In the first round, we held 5 races (with 5 horses in each race), so that all horses participate in exactly one race.
In the second round, we have 1 race: only those horses participate, who won a race in the first round (there are 5 of them). After this race, we have the fastest horse overall (it took 6 races so far).
Let's call the group of horses, who were in the first round in the same race and includes the horse, who:
• finished 1st in the second round: A
• finished 2nd in the second round: B
• finished 3rd in the second round: C
• finished 4th in the second round: D
• finished 5th in the second round: E
Moreover, in order to identify each horse, let's add the place they achieved in the first round to our notation, so A1 will be the fastest horse overall, C4 will be the horse, who finished 4th in the first round in the race where its winner finished 3rd in the second round and so on.
Now, it is easy to see that:
I. We can eliminate the groups D and E entirely and leave just C1 from group C, if we want to determine the 2nd and 3rd fastest horse overall, because there are at least 3 horses, who are faster than any of them: A1, B1 and C1.
II. Horses A4 and A5 cannot be in the top 3 either (as A1, A2 and A3 are all faster than them) and we neither have to consider horses B3, B4 and B5 (as A1, B1 and B2 are all faster than them).
III. As we already know, A1 is the fastest horse overall, therefore we don't have to include A1 in any further race(s) to determine the 2nd and 3rd fastest.
At this point, we are left with 5 horses (A2, A3, B1, B2 and C1).
So in the last round, we have only 1 race: the winner will be the 2nd fastest overall and the runner-up will be the 3rd fastest horse overall.
Hence, 7 races are enough to determine the top 3 horses (in terms of speed).
To see, that 6 or less races are not sufficient to determine the top 3 (so 7 is the minimum), we can consider this:
• at least 25 ÷ 5 = 5 races are needed to give a chance to each horse to race
• however, as we do not know the relative speeds amongst the horses in different groups, we have to consider all the winners as potential overall fastest and we have to have the 6th race
• in the worst case, one first round group (A or B) might contain more than one of the top 3 horses, so we need the 7th race.
Therefore, our answer should be: 7