At the county fair a vendor has 25 horses for sale. He assures us that everyone is very fast and that none runs at the same speed as the others. We want to buy three horses to participate in the races of the racecourse and the seller allows us to run them in races of a maximum of 5 horses each to find the fastest. We do not have a stopwatch so the only way we have to determine the three fastest is by running them between them since they assure us that one horse faster than another will always win a race in which both horses participate.
How many races do we need to do at most to find the 3 fastest horses?
We run the 25 horses in groups of 5 which gives us a total of 5 races with which we will determine the 5 fastest horses in each group.
We made a second round with a race between the five horses that won their respective first races. However, it would be possible for a horse of those who were in second position in the first phase of races to be faster than one of another group that came first in those races so we run the first three horses of the second round with those who took second place in the races in which they did not participate. The first three horses of this third round will be the fastest three.
In total we will have needed 7 races.