Problem of the Month for February, 2006

Problem:

In an arm-wrestling tournament, each person competed against each of the other competitors exactly once. Each pair of competitors had 30 seconds to win the match. Each participant scored 1 point for a win, -1 point for a loss, and 0 points for a tie. Bobby finished with 5 points and Cindy finished with 8 points. Was there definitely a tie at some point in the tournament? Either explain why there was definitely a tie at some point in the tournament or give an example of a tournament where there was no tie (and Bobby and Cindy had the given number of points).

Hint:

The number of competitors is unknown, but you can still deduce things about the number of competitors if you make assumptions about whether there were any ties or not.

Generalizations to consider:

What can you say about a game where Bobby finishes with an odd number of points and Cindy finishes with an even number? How about if they both get odd numbers of points? How about if they both get even numbers of points? (If you can't get answers to the general cases, try special cases).