Winning team
Problem
Nine cross country runners compete in a team competition in which there are three matches.
In the first match team X races against team Y, in the second it is team Y against team Z and in the third it is team Z against team X.
The runners are A, B, C, D, E, F, G, H and I and in all the matches, when they run, they finish in this order.
The teams are team X with runners A, F and H; team Y with runners B, D and I; and team Z with runners C, E and G.
In each match all the runners finish the course and 21 points are awarded, 6 points to the first, 5 to the second, 4 to the third and so on.
The judges work out the results of each of the three matches and try to decide on the winning team for the competition overall.
If you were a judge what would you decide and what is peculiar about the mathematics involved in this?
Student Solutions
Helen, Lucy, Rosy, Becki, Jo and Morven, from The Mount School, York and Henry of St Peter's School, Adelaide all found that the three teams scored an equal number of points so that deciding on the winning team presents difficulties. James of Hethersett High School, Norwich gave the race results as follows.
(1) X v Y
| Teams | X | Y | Y | X | X | Y |
| Runners | A | B | D | F | H | I |
| Points | 6 | 5 | 4 | 3 | 2 | 1 |
Team X 11 points, team Y 10 points so X beats Y.
(2) Y v Z
| Teams | Y | Z | Y | Z | Z | Y |
| Runners | B | C | D | E | G | I |
| Points | 6 | 5 | 4 | 3 | 2 | 1 |
Team Y 11 points, team Z 10 points so Y beats Z.
(3) X v Z
| Teams | X | Z | Z | X | Z | X |
| Runners | A | C | E | F | G | H |
| Points | 6 | 5 | 4 | 3 | 2 | 1 |
Team X 10 points, team Z 11 points so Z beats X.
Adding the points for the races together each team scores 21 points.
These results are very peculiar because each team scored an equal number of points. This means that the judge has to look at which team came first in the three races. As team X beat Y in the first race and team Y beat Z in the second you would expect that, with the same runners finishing in the same order, X must surely beat Z in the third race. This does not happen so you cannot decide on a winning team. You might say X are the winning team because they have the champion runner in their team, but then Z could complain because they beat X.
Compare this idea to the relation of `greater than' for numbers. For example if x, y and z are numbers and we know that x > y and y > z then it must follow that x > z. It follows because x is to the right of y on the number line and y is to the right of z. This relation is called `transitivity' in mathematics and we come to expect it, so when a relation arises that is not transitive, as in this example, it comes as a surprise. What seems obvious is not always true, so when you think you have a mathematical result you might be wrong. You must always prove a result before you can be sure it is true.