Contest Horse Race Puzzle

stahrwe · Mon Jan 10, 2011 9:17 am

This recently appeared in a list of odd questions asked of job interviewees. You may google it if you wish, but I found many different answers all with logical arguments in their favor.

There are 25 horses entered into an afternoon of racing. Each race consists of five horses. How many races must be run in order to determine the fastes horse? You do not have a time keeping device.

Interbane · Mon Jan 10, 2011 10:32 am

stahrwe · Mon Jan 10, 2011 11:20 am


	Interbane wrote: Six?

Nice, now let's complicate it a bit:

Out of 25 horses, pick the fastest 3 horses. In each race, only 5 horses can run at the same time. What is the minimum number of races required?

stahrwe · Mon Jan 10, 2011 7:20 pm

I want to encourage eveyone to take a shot at this. I saw this online as a question asked candiates during interviews. What makes this so fun is that I also read the numerous answers to the question. I'm not sure there actually is a right answer to don't feel intimidated if you aren't a math whiz. Some of the answers had ingenious explanations and they were the most interesting ones. One definitely wasn't appropriate for horse lovers. Essentially you end up with three horses left. I don't recommend that but let your imagination run.

Interbane · Mon Jan 10, 2011 8:49 pm

Eight.

stahrwe · Tue Jan 11, 2011 12:47 pm

Here is a rationale for 7


	Isac Costa wrote: Run 5 races, let's say, Hij denoting the j-th rose in the i-th race, 1 <= i <= 5, 1 <= j <= 5. Eliminate the last two, since there will be three or more horses faster than them for sure. We are left with 15 horses. Run a 6th race with the Hi1 (the 1st placed horses). Suppose the results are Ha1, Hb1, Hc1, Hd1 and He1. The fastest horse in this race (Ha1) is the fastest horse of them all. The last two (Hd1 and He1) can also be eliminated because there will be three or more horses faster than them for sure (at least the top 3 of the 6th race). At this point, we have 12 horses to consider. But the 2nd and 3rd places horses in the preliminary races of the two worst placed horses (Hd1, He1) in the 6th races (Hd2, Hd3, He2, He3) can also be eliminated since there will be at least three horses faster than them. We have only 8 horses left. What about the 2nd and 3rd places horses in the preliminary race of the 3rd placed horse (Hc1) of the 6th race? There will be at least three horses faster than them (Ha1, Hb1 and Hc1). They can be eliminated. We have 6 horses left. Following the same reasoning, the 3rd placed horse in the preliminary race of the 2nd placed horse (Hb1) of the 6th race can be eliminated because there will be at least three horses faster (Ha1, Hb1 and Hb2). We have 5 horses left, namely Ha2, Ha3, Hb1, Hb2 and Hc1. These horses will run in the 7th race and the top 2 will join Ha1 and they are the top 3 horses among the 25 initial. Note that the maximum numbers of races required is MAX = 1 + (H - R)/(R - T), if you have H horses, if you can use R horses per race and if you want the top T horses among all H. For H = 25, R = 5 and T = 3, MAX = 11 races. You get to this result simply through successive races eliminating the last two (R - T) and adding the same number of horses until all of them have raced. It is an exhaustive "brute force" approach.

Interbane · Tue Jan 11, 2011 7:29 pm

Thanks, that's a fun problem.

stahrwe · Tue Jan 11, 2011 9:57 pm


	Interbane wrote: Thanks, that's a fun problem.

Based on that argument do you still think 8 is correct or is 7 right or a different number?

Interbane · Wed Jan 12, 2011 12:07 pm

Seven is correct, by process of eliminating the slowest. The quote you posted explained it, though it was worded sloppily.

earhartam · Fri Jan 21, 2011 7:08 pm

Not sure if the puzzle was actually solved officially, but I'm new and I'd like to add my "two cents." :-)

I still think the answer is six. If six is the correct answer to question 1 (and I think it is) which concluded that the fastest horse can be determined by the five winners of the first five races competing against each other for race number 6, then it stands to reason that the sixth race can also determine the top three fastest horses. If the fastest of the 25 horses can be determined without "him" having to race the other horses from the other 4 races, then there is no need for any additional races to determine the top 3 fastest horses.

Suzanne · Sat Jan 22, 2011 1:30 am

Is there a smiley for; wow, that went way over my head?

Hello and welcome earhartam!

stahrwe · Sat Jan 22, 2011 1:30 pm


	earhartam wrote: Not sure if the puzzle was actually solved officially, but I'm new and I'd like to add my "two cents." I still think the answer is six. If six is the correct answer to question 1 (and I think it is) which concluded that the fastest horse can be determined by the five winners of the first five races competing against each other for race number 6, then it stands to reason that the sixth race can also determine the top three fastest horses. If the fastest of the 25 horses can be determined without "him" having to race the other horses from the other 4 races, then there is no need for any additional races to determine the top 3 fastest horses.

I intended this puzzle to be open ended. I have seen numerous answers with their own reasoning so while ther is an optimal mathematical answer, I don't think there is a best answer in that the reasoning is part of the process.

There is actually a way to determine the fastest three horses without a single race but that is way out of the box thinking and not a palatable one.

Thanks for your input.

earhartam · Sat Jan 22, 2011 3:23 pm

You'll have to trust when I say my thinking is never inside the box, sad to say, I would love to know what it looks like in there. My answer is based solely on answer being six. Otherwise, if we are looking for another answer, the origianl hypothesis, the original equation needs to be amended. So answer to the question as determined by the participants stands until the first answer is alterered.

Thanks for letting me play! This was fun :-)

Forum rules

Contest Horse Race Puzzle Tweet This Topic

Who is online

Chat Room Always Open!