Enigma 1398: Not the World Cup
25 August 2013
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From New Scientist #2558, 1st July 2006
Not being a football fan, I have been following a different international competition. The teams entering are divided into groups of equal sizes and then, within its own group, each team plays each other team once. Then just the two top teams from each group qualify and proceed to the next round, the rest being eliminated. The qualifying teams then have a knock-out competition (with some “byes” through the first round if necessary) eventually progressing to the semi-finals and then the final. Also, the two losing semi-finalists play to decide the third and fourth places.
Having counted the number of teams entering this year, the groups have been decided and the competition is about to start. In the end the average number of games played by each team will be a whole number and no team can possibly play precisely six games.
How many teams entered the competition?