# Number Of races needed?

25 horses, 5 race tracks; we need to choose the top 3 horses. What is the minimum number of races you need?

I have thinking deeply and figure out this: First race all 25 horses in groups of 5 to figure out the top 3 in each group (which will leave 15 horses). Call them: A1 B1 C1 D1 E1 A2 B2 C2 D2 E2 A3 B3 C3 D3 E3 Here, A1 is faster than A2. A2 is faster than A3. The same applies for the rest of the groups. Now race A1, B1 C1 D1 E1 Lets say the order of the horses according to ranking was A1, B1,C1,D1,E1

So A1 is No 1. Now A1 is faster than B1 and B1 is faster than C1. So we can get rid of the entire D and E groups. 9 horse remain. Also, A1 is faster than B1 and B1 is faster than C1. So we can get rid of C2 and C3. Now 7 horses remain. A1 is faster than B1. B1 is faster than B2. Get rid of B3, and 6 horses remain.

Out of these, we know A1 is the fastest. So now race A2, A3, B1,B2,C1 to figure out No 2 and No 3 positions.

-
What is the question? You seem to have a solution and to understand it. –  Ross Millikan Aug 7 '11 at 18:01
I just want to find out whether my solution is wrong or right? –  rohit Aug 7 '11 at 18:07
I agree with it. –  Ross Millikan Aug 7 '11 at 18:08
@Rohit: Please don't use text abbreviations like "u"; given that you are writing in a math site, it's too easy to misread or be confused. In any case, you aren't paying by the letter. –  Arturo Magidin Aug 7 '11 at 18:28
Why can't you just run them all off against each other??? i.e. one race. –  Jp McCarthy May 21 at 10:36

Let's take an assumption, each horse runs at uniform speed every time. They all have different but fixed speed tag.

Step 1) Divide the 25 horses into 5 groups. Each group will have 5 horses each.

Step 2) Conduct 5 races for the 5 groups.

Step 3) Conduct an another race with each group winners. (i.e. Fastest from Group 1, Fastest from Group 2, ... , and Fastest from Group 5)

Step 4) Choose the winner from the Step 3. That will be fastest horse. (So, it needs only 6 races to find out the fastest horse of all the 25 horses)

Additionally, Step 5) Name the horses as below: {a1,a2,..., a5}--> For the group which has the fastest horse after Step 3 (6th Race) {b1,b2,...,b5}---> For the group which has the 2nd fastest horse after Step 3 {c1,c2,...,c5}----> For the group which has the 3rd fastest horse after Step 3 {d1,d2,...,d5}---> For the group which has the 4th fastest horse after Step 3 {e1,e2,...,e5}---> For the group which has the 5th fastest horse after Step 3

Where according to their speed, a1>a2>..>a5; b1>b2>..>b5; c1>c2>..>c5; d1>d2>..>d5 and e1>e2>..>e5, also, a1>b1>c1>d1>e1, but not necessarily, b1>a2, c1>b2 etc. [Clearly a1 is the fastest horse]

Step 6) Conduct a race with a2, a3, b1, b2 and c1 to find out the 2nd fastest and 3rd fastest of all the 25 horses. So, we need to conduct only 7 races to find out the top 3 fastest horses!

-
I think that solution has flaw that he assume 1st winner by taking time of all 5 races if we can assume winner of 25 horses by taking time of all horses why not 2 and 3rd winner. –  rohit Aug 7 '11 at 18:05
@rohit: it appears you are not allowed to time the horses, you must race them to put them in order. So after the 6th race you have established the fastest and have only five left that can possibly be 2nd and 3rd. This allows the 7th race to find those two. –  Ross Millikan Aug 7 '11 at 18:09
yes this what i am thinking i am just want to confirm my answer –  rohit Aug 7 '11 at 18:22
Though I am not downvoting this, I do want to politely register my disapproval of answers like this. What happens if the link breaks in the future? When composing an answer, please consider the long-term value of what you place in the answer box. –  cardinal Aug 7 '11 at 20:00
I am curious: can it be proved rigorously that 7 is the minimum number of races required? Does it belongs to (or relates to) some active mathematical research area? –  minimax Mar 7 '14 at 20:56

Seven races:

1. Five heats of 5 horses
2. A final of the 5 winners, to crown the fastest horse. Retire the horses that finished fourth and fifth.
3. A playoff, with the remaining horses from the final, as well as the horses from the heats that finished second and third to the champion, and second to the runner-up. The champion doesn't need to be in the playoff. The top two horses in the playoff are the fastest after the champion.
-