Mathematics and Sports
Posted April 2010.
Some of the fascinating mathematics of sports scheduling...
Sports in America is both big business and democracy in action. Baseball, football, and basketball have become writ large, with games on television and cable television generating huge sums of money that goes into the American economy. But sports are also a mechanism whereby American children learn the values of fair play, hard effort, and the meaning of friendship.
Graph theory, a branch of combinatorics which draws heavily on geometrical ideas, uses diagrams consisting of dots and lines to help get insight into a variety of mathematical problems. The complete graph on n vertices has exactly one edge between every pair of vertices. These graphs are denoted Kn; Figure 1 shows K4 and Figure 2 shows K5.
We can use the graph below to visualize what is going on. The edge 1 to 6 is shown vertically in a diagram where the vertex 1 is placed at the "center" of a regular pentagon and the numbers 2 and 3 are listed clockwise starting from 1, while the numbers 5 and 4 are shown counterclockwise starting at 6. In the diagram shown the edges which make up the sides of the regular convex pentagon are omitted. Only the edges which make up the pairings in one round for the teams are shown. The other edges of the matching (in addition to the pairing of 1 and 6, shown in red) are shown horizontally in blue. (Two colors are used to highlight the different role of the vertical and horizontal edges in the diagram. However, in the partition of the complete graph on 6 vertices into 3 disjoint edges, these three edges would be in one color class.) Since there are 5 rounds each with 3 edges we can account for all of the 15 edges in the complete graph on 6 vertices.
Here is an analogue of the diagram above after the rotation of the vertices in the regular pentagon:
with a corresponding labeled graph:
and a graph-theoretical way to see what is going on:
One additional sample of the rotation in the visual graph theory environment:
Extensions and generalizations
One natural pragmatic concern for scheduling tournaments is where the games are played. For some tournaments the games may be played on "neutral" territory where the opponents are not at some advantage because of being on their home field or having the encouragement of the hometown fans. However, in a lot of sports there is this issue of home and away games. Is there some easy way to take this issue into account?
In some situations the issue of home/away does not matter but it usually does. Thus, for each team, one can produce a sequence of n-1 H's and A's which represent the home/away pattern of games that team must play.
When you sit down to watch your favorite sports star or team I hope you will recognize the behind-the-scenes role that mathematics is playing in bringing these events to you and making it possible to have fair, competitive and efficient sports events. If you are a "little league" mom or dad or participant, perhaps you can enjoy the mathematics behind the sports, as well as the sport itself.
Anderson, I., Constructing tournament designs, Mathematical Gazette, 73 (1989) 284-292.
Those who can access JSTOR can find some of the papers mentioned above there. For those with access, the American Mathematical Society's MathSciNet can be used to get additional bibliographic information and reviews of some these materials. Some of the items above can be accessed via the ACM Portal , which also provides bibliographic services.
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These web essays are designed for those who have already discovered the joys of mathematics as well as for those who may be uncomfortable with mathematics.
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