There are 25 horses and you need to figure out the 3 fastest horses by placing them into races. You can race 5 horses at a time. Each horse always finishes the race in the same amount of time and there are no ties. The only information you get from each race is the order that the 5 horses finished in. You will not get any information regarding the time taken for the horses to complete the race.
What is the smallest number of races you need to find the 3 fastest horses in order?
7 races.
Let’s name the races R1 through R7. Let Rxn represent a horse in race x, finishing in nth place. So R32 represents a horse that finished 2nd place in the 3rd race.
Group the 25 horses into 5 groups of 5 and race them. The 4th and 5th placed horses of each race can be eliminated since they cannot meet the criteria of 3 fastest horses. We are now left with 15 horses (5 groups of 3 horses); 3 horses from each race.
For the 6th race, race the fastest horse (R11, R21, R31, R41, R51) from each of the first 5 races. The winner of the 6th race is the fastest horse. The 4th and 5th placed horses from the 6th race can be eliminated including all the horses within their respective groups. For example, if the horse that came in 4th place is from R41, the horses R42 and R43 can be eliminated as well.
Let’s say for the 6th race, R11 came first, R21 came 2nd and R31 came 3rd. We know that R11 is the fastest horse. We now need to determine the 2nd and 3rd fastest horses.
We can now also eliminate the horses R23, R32 and R33. This will leave us with five horses for the 7th race – R12, R13, R21, R22 and R31.
The 1st and 2nd placed horses in the 7th race are the 2nd and 3rd fastest horses.