Horse Racing

Given a horse race with 5 gates and no stopwatch find the 1st-3rd place winners of a pool of 25 horses  assuming that each horse will race consistently.  How many and what races must be run in order to determine the winners ?

SOLUTION:

This question is an interesting one. Let’s start by asking ourselves, what are our limitations? Well, we don’t have a stopwatch so we can’t time the horses. That means that we can’t compare the race times of each horse – otherwise we could simply have 5 races of 5 horse each and pick out the 3 fastest times to get the 3 fastest horses. Remember that we can only race 5 horses in each race otherwise we could just have one race with 25 horses, pick the 3 fastest, and we would be done!!!.

We are going to use matrix approach to visualize this problem.First divide 25 horses into 5 groups to get a 5×5 matrixby putting each horse randomly into a 5×5 matrix.where each row represent a race and each column represent the rank in the corresponding race.

1)First  5 races will give the winner of each race and the order(rank) in which they should appear in the matrix.Now we will get the matrix where each row represent a race and each column represent the rank in the corresponding race e.g.  horse X_2.3 got the third rank in the second race.Note that the 10 horses X_?.4-X_?.5 were eliminated in the first five 5 races by virtue that they cannot place 1^st-3^rd.

X1.1	X1.2	X1.3	X1.4	X1.5
X2.1	X2.2	X2.3	X2.4	X2.5
X3.1	X3.2	X3.3	X3.4	X3.5
X4.1	X4.2	X4.3	X4.4	X4.5
X5.1	X5.2	X5.3	X5.4	X5.5

2)A winner of 6^th race of  the individual winners ( X_?.1 i.e. first column) of the first 5 races will give the clear 1st place winner(Suppose here X_1.1) and suppose if  the 2nd and 3rd winner of this race is X_2.1 and X_3.1 then the remainder of the last two rows can be eliminated along with the horses labeled X_3.3, X_3.2, X_2.3 as they will not be able to place 2^nd or 3^rd either.

X1.1	X1.2	X1.3	X1.4	X1.5
X2.1	X2.2	X2.3	X2.4	X2.5
X3.1	X3.2	X3.3	X3.4	X3.5
X4.1	X4.2	X4.3	X4.4	X4.5
X5.1	X5.2	X5.3	X5.4	X5.5

3) The 7^th and last race will be between non-eliminated horses labeled as  X_1.2, X_1.3, X_2.1, X_2.2, X_3.1 which will give  2^nd and 3^rd place winner.

So a total of 7 races of 5 horses each is required to find the 1^st-3^rd place winners…Was it tough!!! 😉

If that was not enough for you ……

If you want to determine a fourth place winner, then you need at most only one more race.

• If  X_1.2 wins and X_1.3 and X_3.1 share 4^th or 5^th places in the last race, then  X_1.2,  X_2.1 and X_2.2 take 2^nd, 3^rd and 4^th places respectively overall without another race.
• If in the last race, X_1.2 and X_1.3 take 4^th or 5^th places respectively and X_3.3 takes 2nd then X_1.4, X_2.2 and X_3.2 must race to determine the 4^th place overall.
  
• Otherwise, the final races 3^rd place winner must face off against the neighbor to the final races 2^nd place winner for 4^th place overall