In an article on the recent National Masters XC race I neglected to mention one important and unusual feature of the race: scoring was done by totaling the times of the scoring members on each team. Lowest total time wins. (There were varying numbers of "scoring" members in the various age divisions. For example, the first 5 finishers' times were totaled in the Men's 40-49 division, but only the first 3 finishers' times were used in the Men's 70-79 division.) Why is this worthy of note? The normal method used to score XC races is to total the *places* of finishers rather than their times. One effect of the usual method is deemphasize the contributions of very fast runners (and the failings of very slow ones.) The score is more reflective of team than individual preformance, and seems more suitable to XC -- it being, after all, the quintessential team sport. I wonder why USATF has chosen the unusual total time method? I'm sure they must have a reason, but it escapes me. While I have the floor, I can't help rambling on a bit more about the usual XC scoring, as it exhibits some interesting features from a mathematical point of view. Consider, e.g, a 3-way meet involving teams we will call (unimaginativly) A, B, and C. In addition to the usual 3-way competition, it is not unusual for such meets to function as dual meets between each of the 3 possible pairs of teams. It may seem counterintuitive, but it is possible for team A to beat team B, team B to beat team C, and team C to beat team A! This holds in the following example; moreover, the margin of C's victory over A is more decisive than in any other pairing. Example. Suppose for simplicity that 3 runners on each team score and that the runners finish as shown: Place: 1 2 3 4 5 6 7 8 9 Runner: A B C B C C A A B ( Only the team affiliation of a runner is recorded. ) In the 3-way meet, Team C wins with 14 points, team B is second with 15 points, and team A is third with 16 points. Now consider the A vs B dual meet. Runners for team C are irrelevant for this meet, and so are ignored. We have Place: 1 2 3 4 5 6 Runner: A B B A A B, and team A beats team B 10 points to 11. Similarly, it is easy to check that team B beats team C 10 points to 11, and team C beats team A 9 points to 12. This phenomenon is a well known and serious issue in the mathematics of voting. The interested reader may wish to consult the following monograph for further discussion: D. Saari, Basic Geometry of Voting, Springer- Verlag, 1995. Added, 11/25/2010: Recently the question arose whether it is possible to have a tie score in a cross country meet between two teams. In the most common scenario, where 5 runners on each team (and no others) count toward the score, the answer is certainly "no". The sum of all ten places is 55, and 55 is an odd number. It is also common, however, to have one or more additional runners on each team who are allowed to influence the score indirectly, by "displacement." While such a runner's own place is not added to the total of his team's, his presence may add to the opposing team's total by increasing the finish place of some of its scoring 5. A common setup is to allow up to 2 displacing runners. In this case, with 5 scorers, it is possible to have a tie. In the following example, each column gives the finish places of up to seven runners from one team, while only the lowest 5 places count in the totals: TEAM A TEAM B -------------- 1 2 3 4 5 6 7 8 9 10 11 12 ---------- 29 29 -- ************************************************************************ Terry R. McConnell Mathematics/304B Carnegie/Syracuse, N.Y. 13244-1150 trmcconn@syr.edu http://barnyard.syr.edu/~tmc ************************************************************************