Find min races required: 25 horses, 5 tracks, 3 fastest horses puzzle

Yogesh

8 years ago

25 horses, 5 tracks & 3 fastest horses puzzle

Given 25 horses and a racing track.
- 5 horses can run simultaneously on a racing track.
We want to find 3 fastest horses among 25 horses.
Find the minimum number of races required to find 3 fastest horses.

Let us first organize the horses in five groups and each group having five horses.
We will carried out five races among the groups and results are as shown in Table 1

Table 1 : Races of five groups
Group	Horses	Race Result	Top 3 horses
1	H1, H2, H3, H4, H5	H1, H2, H3, H4, H5	H1, H2, H3
2	H6, H7, H8, H9, H10	H6, H7, H8, H9, H10	H6, H7, H8
3	H11, H12, H13, H14, H15	H11, H12, H13, H14, H15	H11, H12, H13
4	H16, H17, H18, H19, H20	H16, H17, H18, H19, H20	H16, H17, H18
5	H21, H22, H23, H24, H25	H21, H22, H23, H24, H25	H21, H22, H23

We got five fastest horse among five groups.
We will arrange 6th race among the winners.
- H1, H6, H11, H16, H21 and results are shown in Table 2.

Table 2 : Sixth Race result
Group	Horses	Race Result	Top 3 horses
6	H1, H6, H11, H16, H21	H1, H11, H21, H6, H16	H1, H11, H21

Let us analyze the race result of Group 6 (Table 2)
- We got H1 as winner among the fastest horses of group 1 to 6. (Table 1 and 2)
  - H1 is the fastest horse among 25 horses.
  - Now, we will find horse for 2nd and 3rd place.
- H6 is fourth in group 6 Race.
  - H6 can not be among top 3 fastest horses.
  - We will eliminate H6 horse.
  - H6 was winner in Group 2 race
    - Group 2 horse are slower the H6 horse.
    - We can eliminates all horses of group 2.
- H16 is fifth in group 6 Race.
  - Similarly, we can eliminates group 4 horses.
We have summarized results in Table 3
- We have eliminated group 2 and group 6 from table 1..
- We have shown Group 6 result (for reference).
- Now, We will perform the elimination from Table 3 as per last six races.

H21 was third as a result of Group 6 result.
- In group 5, H22 and H23 can not be top 3 horse. (eliminate them)
- In group 5, We will have H21 which can be among top 3 horses.
H11 was second in Group 6 result.
- In group 3, We can take one horse which can come third.
- We can eliminate the slowest horse from group 3 i.e. H13.
- In group 3, we can have H11, H12, which can be among top 3 horses.
H1 was first in Group 6 result.
- In group 1, H2 and H3 can come among top 3 horses.
Finally we will have, H2, H3, H11, H12, H21 horse which can be among top 3 horses.
We will arrange 7th race to find second and third fastest horse.
- Suppose result of 7th race is H2, H12, H11, H2, H21
Top 3 horses among 25 horses in 7 races are H1, H2 and H12.