Problem of the Week 1184

World Cup Groups

In soccer's World Cup, a group consists of four teams, each playing the other three once. A win is worth 3 points, a draw gives 1 point to each team, and a loss is worthless. After the 6 games, there is a vector (a, b, c, d) where a is the number of points obtained by the top team, b the points of the second team (possibly b=a), and so on. Each of a, b, c, d is an integer between 0 and 9, though of course a is at least 3 and the sum is between 12 and 18. How many distinct vectors (a,b,c,d) can arise?

Further thoughts: Assume p to be the probability that any game ends in a draw and equal probability of either team winning in the 1-p case that a game ends in a win. What is the most likely vector? The least likely? A reasonable value of p might be 1/5.

Real world query: Has it every happened in a world cup (or other major tournament with the same format) that all 6 games were draws, so that the points were (3,3,3,3)?

© Copyright 2014 Stan Wagon. Reproduced with permission.

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25 June 2014