A little logic puzzle

S Aufrecht

Here’s a little logic puzzle for you.

Four football teams form a set. Each team plays the other three teams once each. The two top teams advance to the next round. A victory is worth three points, a tie one, a loss none. Should two or more teams be tied on points, the winner by the first applicable method from this list (p17):

  1. greater number of points obtained in the group matches between the teams concerned;
  2. goal difference resulting from the group matches between the teams concerned;
  3. greater number of goals scored in the group matches between the teams concerned;
  4. goal difference in all the group matches;
  5. greater number of goals scored in all the group matches;
  6. drawing lots by the Organising Committee for the FIFA World Cup™.

For convenience, let’s call the teams Italy, Ghana, the Czech Republic, and the United States. And let’s say, for the sake of the logic puzzle, that each team has played twice, with the following results:

Under what conditions will the US team advance?

Answer: The US team will advance if it beats Ghana and one of the following is true:

Hypothetically speaking.

Addendum: both Yahoo Sports and ABC TV indicate that the US could advance if Italy ties Czech and the US beats Ghana by enough goals. They must be looking at some different rules than I saw. If Czech ties Italy, the US and Czech are tied for second in the group with four points each. Then it goes to “greater number of points obtained in the group matches between the teams concerned.” In the only group match between the US and Czech, Czech got three points and the US zero. Goal differential doesn’t even come into play. End of story.

Further Addendum 20 June: It turns out that there are indeed two different sets of rules up on the FIFA website. This page, which is labelled “Preliminaries”, has Regulations 2006 FIFA World Cup Germany™, which were ratified in March 2003. But this page, labelled “Tournament” but otherwise identical in appearance, has Regulations 2006 FIFA World Cup Germany™, ratified in December 2005. The tie-breaking rules were changed between the two versions. The apparent reason is that “having b) as the first tie-breaking criterion avoids the absurd situation … in group G, where even if France wins 10-0 against Togo, if Switzerland and Korea draw 2-2, Switzerland and Korea make it through. […] It is the same situation that saw Italy off in euro 2004, when Sweden and Denmark needed a 2-2 to go through and they did. Their supporters came to the game with “2-2″ banners and it was just ridiculous….” —Mircia, commenter, New York Times World Cup Blog. The analysis above is correct for the 2003 rules, but the rules now in effect are:

The ranking of each team in each group will be determined as follows:
a) greatest number of points obtained in all group matches;
b) goal difference in all group matches;
c) greatest number of goals scored in all group matches.

If two or more teams are equal on the basis of the above three criteria, their rankings will be determined as follows:

d) greatest number of points obtained in the group matches between the teams concerned;
e) goal difference resulting from the group matches between the teams concerned;
f) greater number of goals scored in all group matches between the teams concerned;
g) drawing of lots by the Organising Committee for the FIFA World Cup™.

Under these rules, the US can advance even if Italy and Czech tie; see this comment for an updated analysis.