Suppose you have 25 horses, and you want to pick the fastest 3 horses out of those 25. In each race, only 5 horses can run at the same time in a race because there are only 5 tracks. What is the minimum number of races required to find the 3 fastest horses without using a stopwatch?
Have visual representation
| a1 | a2 | a3 | a4 | a5 |
| b1 | b2 | b3 | b4 | b5 |
| c1 | c2 | c3 | c4 | c5 |
| d1 | d2 | d3 | d4 | d5 |
| e1 | e2 | e3 | e4 | e5 |
In each row, the fastest horses are listed in descending order, from the fastest (extreme left) to the slowest (extreme right).so in the first race a1 was the fastest and a5 was the slowest. In the second race b1 was the fastest, b2 was the second fastest and so on.
Next :
we’ve had 5 different races, we can eliminate the slowest 2 horses in each group since those horses are definitely not in the top 3 . so we have now matrix as a1,a2,a3 ..to.. e1,e2,e3.
Next :
we can have 6th race with a1, b1, c1, d1, e1 so that we can have fast 3 .
(Why to do so..
the horse say e1 and d1 came 4th and 5th are not fastest 3 but together we can eliminate d2,d3 and e2,e3 as they are also slower than d1 and e1 and so they are slower then a1,b1,c1 )
Hence finally we have
a1 , a2 , a3
b1 , b2 , b3
c1 , c2 , c3
Now as a1 is the fastest among fastest, we can say the first fastest is a1 . and we don’t need to keep a1 in race to get second and third fastest.
we have
a2, a3
b1 , b2 , b3
c1 , c2 , c3
Which horses still we can eliminate from sixth race.
a1 is fastest, for second and third position b1,c1, b2 can only be possible candidate.
as b3 is slowest in b1,b2,b3 so it can’t take 2nd, or 3rd fastest. similarly c3 can’t be in 2nd and 3rd fastest.
Now as c2 is slower then c1 and c1 is slower then b1. (in sixth race a1 was first, b1 was second, and c1 was third)
hence c2 also can’t be in 2nd and 3rd fastest.
Finally we have
a2, a3
b1 , b2
c1.
Have 7th race among this 5 horse . get fastest two
say b1 and a2. (or any)
We have top three fastest a1,b1,a2.