Determine Fastest 3 horses out of 125 when only 5 racing track are given without using stopwatch? We have 125 horses, and we want to pick the fastest 3 horses out of those 125. In each race, only 5 horses can run at the same time because there are only 5 tracks. What is the minimum number of races required to find the 3 fastest horses without using a stopwatch?
This question was easily solved when there were 25 horses but for 125 horses it is not solvable easily.Can anyone help me out with this complex problem?
This is the link of the question which is also not solved.
 A: The answer according to me should be $33$.
EXPLANATION: 
Let us mark the horses as $X_i$ where $i=1,2,3,..,125$ 
First we divide the $125$ horses into $25$ groups of $5$ each. 
Say we divide the horses in such a way that the $n^{\text{th}}$ group contains horses $X_{5n-4}$ to $X_{5n}$. 
For each group, there is a group race where all the $5$ horses of the group run and the position-holders are separately identified. 
In this way, $25$ group races are held.
Without loss of generality and solely for our purpose to explain the problem in easy language, we assume that the $1^{\text{st}}$, $2^{\text{nd}}$, $3^{\text{rd}}$, $4^{\text{th}}$ and $5^{\text{th}}$ position holders in the $n^{\text{th}}$ group race are respectively the horses $X_{5n-4},X_{5n-3},X_{5n-2},X_{5n-1}$ and $X_{5n}$.
Next we take the winners of the $25$ group races, whom we call the finalists.
From these $25$ horses, the fastest $3$ can be determined by holding $7$ races as per the 25 horse race problem. 
So, by now, to keep track of the number of races, $25+7=32$ races are held.
Finally, again without loss of generality and solely for our purpose to explain the problem in easy language, we assume that the $1^{\text{st}}$, $2^{\text{nd}}$ and $3^{\text{rd}}$ position holders of this final race are $X_1,X_6$ and $X_{11}$ respectively. 
Now since we are interested only in the three fastest horses, we can ignore the other $22$ finalists and the horses these $22$ finalists had defeated in their respective group races since none of the finalists can be among the top $3$ and the horses they had defeated also cannot be due to the same reason.
So the situation boils down to the $3$ horses $X_1,X_6$ and $X_{11}$ and the $12$ horses they had defeated in their respective group races i.e. $15$ horses in total.
Next comes an important logic: 

*

*We exclude $X_1$ from all our calculations since we know it is the fastest.

*$X_{11}$ can be at best the third fastest, so the horses it had beaten in its group race cannot be among the top three fastest horses, whatever happens.So we ignore these $4$ horses namely as per consideration, $X_{12},X_{13},X_{14}$ and $X_{15}$.

*Also $X_{6}$ can be at best the second fastest, so among the horses it had beaten in its group race, $X_7$ can be at best the third fastest and the others cannot be among the top three fastest horses, whatever happens.So we ignore those $3$ horses namely as per consideration, $X_{8},X_{9}$ and $X_{10}$.

*Similarly, establishing same arguments, from group $1$, $X_{2}$ can be at best the second fastest and $X_3$ can be at best the third fastest and the others cannot be among the top three fastest horses, whatever happens.So we ignore those $2$ horses namely as per consideration, $X_{4}$ and $X_{5}$.

So from the $15$ horses, we started our logic with, we are ultimately left with the $5$ horses namely $X_{4},X_{5},X_{6},X_{7}$ and $X_{11}$.
We organise $1$ more race with these $5$ horses and determine the second and third fastest horses.
CONCLUSION: 
Thus, we have used in total $25+7+1=33$ races to determine the three fastest horses out of $125$ horses.
Hope this helps.
