FIFA World Cup Group Stage Scenarios By Peter Lowe, M.S. College Instruction: Mathematics, Eastern Washington University The FIFA (Federation International Futbol Association) World Cup is a tournament involving the top soccer teams in the world. It is held every four years, and currently uses a format which involves placing teams into groups of four. Each team must play a match (game) against all three other teams in its group. The team scoring the most goals wins the match with ties resulting in a draw. Teams are then awarded 3 points for a win, 1 point for a draw, and 0 points for a loss. The top two teams from each group (based on points earned) advance to the next stage. This article provides a framework for analyzing the possible occurrences within this type of group play. In a group of four teams the total number of games required for each team to play the others is six. The diagram below illustrates the six games (lines) between the four teams (dots). Each of the games could result either in a win for one team, a win for the other, or a draw. This makes three possibilities for each of six independent events. For one game the total number of possible outcomes would be three. Combining each of those three possibilities with the three possibilities for the second game we have nine different possible ways for two games to finish. As we continue to add games the number of possibilities continues to triple so the total number of permutations for six games would then be 36 = 729. This relatively large number of permutations makes analysis difficult; however, it takes into account a number of permutations that, for the purposes of analysis, are not significantly different. For example, if one team was to beat all three of the other teams, and the other teams were to draw against each other, that would be one possible scenario for the purpose of analysis. (This scenario is illustrated using an arrow to indicate that the team at the beginning of the arrow beat the team to whom the arrow points. Lines without arrows denote draws.) When counting permutations, however, this one scenario would represent 4 of the 729 possibilities because any of the four teams could finish as the dominant team with the others affecting draws indistinguishably. In the same way each of the 729 permutations is not truly unique when analyzing group scenarios because many of these permutations involve simply rearranging the names of the teams achieving certain wins and loses. This means that the 729 permutations can be grouped into scenarios which do not distinguish which of the four teams finishes 1st, 2nd, 3rd, and 4th. This new set of scenarios would be concerned instead with the pattern of wins and losses among the four teams. The task then becomes to illustrate all the group scenarios in such a way that, given the results of any group, one could put the names of the teams on one of these scenarios to illustrate the results of the six games. Determining the number of scenarios possible can be done with a group theoretic approach using the Burside-‐ Frobenius-‐Polya orbit counting formula and some attention to the number of arrangements fixed by the group action. Aside from being somewhat outside of the scope of most recreational mathematicians tools, this would only give us the number of scenarios and we are interested in illustrations of the scenarios themselves.
In finding all the possible scenarios it is important to note that just because a scenario is illustrated differently does not mean that it constitutes a new scenario. The diagram below, for example, looks different from the previously illustrated scenario, but still represents one team beating all three opponents, and those opponents affecting draws with each other. To avoid these duplications we will adopt the convention that the team scoring the most points (#1) will be placed in the center, and the #2, #3, and #4 teams will be placed clockwise starting in the upper left hand corner, so that the team finishing last is placed on the bottom. This does not eliminate the