A Horde Of Horses

Whoever thinks math is plain old boring numbers is missing out on the whole set of ideas that math encompasses. Logical thinking is synonymous with math, and to do this question, we will use just that. How many races would it take to determine the top 3 horses, given 25 horses, wherein each race consists of 5 horses? This does assume that the only information one can obtain from each race is the ordering, and not the times of the horses. This question really makes you think, and is quite involved, so get a piece of paper, and start doing it! The solution lies below the picture, and so scroll only once you’re ready to see it.

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In the start, we will simply have 25 unknown horses, and so the only possible thing that we can do is do 5 races with all the horses. For the purposes of this problem, I will take the letters of the alphabet A-Y to represent the horses. Let’s say that we do five races, and obtain the following results.

1st place 2nd place 3rd place 4th place 5th place
Race 1 A B C D E
Race 2 F G H I J
Race 3 K L M N O
Race 4 P Q R S T
Race 5 U V W X Y

What is it that we want at the end? We want to find out the top three horses, and so any horse that comes 4th or 5th in any race is automatically not needed anymore. So we now come to this.

1st place 2nd place 3rd place 4th place 5th place
Race 1 A B C D E
Race 2 F G H I J
Race 3 K L M N O
Race 4 P Q R S T
Race 5 U V W X Y

Now let’s have a race between the 5 horses which came in first place: A, F, K, P, and U. We have the results of this race here:

1st place 2nd place 3rd place 4th place 5th place
Race 6 K P F A U

Now we have some more observations to make. If A is 4th, we know that it is out of contention, and the same applies for 5th place U. But what we can also eliminate are the horses which are behind A and U in their respect races, as we know they are slower than A and U. The new updated table comes to this:

1st place 2nd place 3rd place 4th place 5th place
Race 1 A B C D E
Race 2 F G H I J
Race 3 K L M N O
Race 4 P Q R S T
Race 5 U V W X Y

We found that among the first place horses, F was third, which means it is still in contention. But what about the horses behind F in Race 2? They have no possibility of getting in the top 3, as they are behind F, and F is already third in at least one race, so they are out. Similarly, we found that P was second in the race of first horses, so the horse directly behind it in Race 4 is possibly third place. But the one behind that is most definitely not in the top 3 (this is 3rd place in Race 4, so R). After eliminating these horses we can see the updated table:

1st place 2nd place 3rd place 4th place 5th place
Race 1 A B C D E
Race 2 F G H I J
Race 3 K L M N O
Race 4 P Q R S T
Race 5 U V W X Y

K won the race between the first place horses, and so is guaranteed to be the overall champion. So K can be left out of any further races. How many horses are we down to now? We only have 5 horses other than K left, and so a race between them will decide the whole ranking:

1st place 2nd place 3rd place 4th place 5th place
Race 7 L P Q M F

We now know that the top 3 ranking of the 25 horses is: K, L, and the P. How many races did we use? We only used 7. This is quite interesting, as we had to do 5 races just at the get-go, and so by only doing 2 races after that we were able to obtain the ranking that we desired. I don’t know how you feel, but something about this problem is extremely satisfying to me.

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