There are 25 horses. At a time only 5 horses can run in the single race. How many minimum races are required to find the top 5 fastest horses? Please explain your answer. (There is no stopwatch to measure the times.)

Question

imqwerty · Answer

1st we form 5 groups each having 5 horses
Lets say the groups are $\color{violet}{G_1,~G_2,~G_3,~G_4,~G_5}$ 

now we organize 5 races between the horses of each group
after the 5 races we get the toppers of each group

total races till now= 5

now we organize a race between all these 5 toppers 
we get to know who is 1st,2nd,3rd,4th and 5th amongst all these 5 group toppers
Lets say that 
$H_1 ~got ~1st$
$H_2~got~2nd$
$H_3~got~3rd$
$H_4~got~4th$
$H_5~got~5th$

the horse which gets the 1st position[$H_1$] in this race is the fastest of all

total races till now=5+1=6

now we take $H_5$ and race it with the horses who got 2nd position in the inter group races [ note that we will not take the horse who stood 2nd in the group in which $H_5$ was because its sure that it will lose]

Now in this race $H_5$ will either get 1st or 2nd or 3rd or 4th or 5th position

$\color{blueviolet}{if ~H_5 ~gets~ 1st ~position}$ then its sure that $H_1,~H_2,~H_3,~H_4,~H_5$ are the top horses.
$total~races~in~this~case=5+1+1=7$


$\color{blueviolet}{if ~H_5~gets~ 2nd ~position}$ then the horse which defeats it will come into the top 5 replacing $H_5$. 
$total~races~in~this~case=5+1+1=7$


$\color{blueviolet}{if ~H_5 ~gets~ 3rd ~position}$ then 2 horses who got 2nd in the group race are ahead of it
Lets call these 2 horses as $H_6,~H_7$ and $H_6~got~1st$ and $H_7~got~2nd$ in this race.
So now because 2 horses i.e., $H_6~and~H_7$ are faster then $H_5$, $H_5$ gets eliminated from the top 5 and we race $H_6~and~H_7$ with the group topper horses[except for the horses who is the group topper of $H_6$'s group, because its sure that he can defeat $H_6$ and $H_7$ both] 
Now we organize the race and we get the top 4 from it and then the other horse who will join the top 5 horse team will be group topper of $H_6$'s group.
$total~races~in~this~case=5+1+1=7$


$\color{blueviolet}{if ~H_5 ~gets~ 4th ~position}$ then there are 3 horses who are ahead of it
lets say they are - $H_6, H_7,~H_8$ where $H_6$->1st, $H_7$->2nd, $H_8$-3rd
NOW, number of horses who are surely ahead of $H_8$-->$H_6,~H_7$ , group topper of $H_6$'s group, group topper of $H_7$'s group and $H_1$, group topper of $H_8$'s group
so no need to consider $H_8$ because 5 horses are already ahead of it
so now we just have to consider $H_6,$ and $H_7$ like we did in previous case when $H_5$ got 3rd position
$total~races~in~this~case=5+1+1=7$



$\color{blueviolet}{if ~H_5 ~gets~ 5th ~position}$here again there is no need to consider $H_8~and~H_9$ and then we just gotta consider $H_6~and~H_7$ like previous case and we know how to do it :)
$total~races~in~this~case=5+1+1=7$

so answer is 7

baru · Answer

if you race H5 against all the horses who came second in the first group race,
then you are racing 6 horses at a time...

imqwerty · Answer

no 
we are not racing the horse who came 2nd in $H_5$'s group 
because its sure that $H_5$ will defeat it and will always be ahead of it.

baru · Answer

oh..yea u mentioned that 
nice solution :)

imqwerty · Answer

:) thanks

baru · Answer

hmm...
lets say by some stroke of luck, G1 happened to have the 5 fastest horses..
how would this play out?

baru · Answer

lets say G1 had the five fastest horses,
then in the group h1, h2 , h3 ,h4 h5,  only h1 belongs

when you race h5 with everyone who came second,  h5 is sure to be defeated by the horse from G1...

if h5 came second, h5 would be replaced by whoever came first( lets say h6)

then in the group for 5 fastest, only two of them belong there...(h1 and h6)

loser66 · Answer

I meant: suppose all 5 fastest horses are on G1, the first 5 races pick out H11, H21, H31, H41 and H51. the 6th race to pick the fastest among the group, surely, it is H11. However,  H12 is the second fastest horse, hence we need another race to eliminate the "fake" fastest among {H21, H31, H41, H51}. Suppose the "LOSER" is H51
Then we need another the race to eliminate another "fake fastest" among {H21, H31, H41}
Hence, we need at least 10 races.