Brainsteem Mathematics Challenges - Drawn Matches
The Question
A football tournament involves 6 teams and each team plays all the other teams once. If a team wins a match it gets 3 points, with the loser getting 0; if a match is drawn then both teams get 1 point. In this particular tournament, it was noticed that every drawn game was between teams with a different number of points on the board at the time of play.
What is the maximum possible number of drawn games?
This is at the level of a secondary national mathematics competition.
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Ok, so we have a round-robin tournament of 6 teams, A through F. There will be 15 matches in all, and we want to generate as many draws as possible. To do that I'll set up the schedule so that only one match is played at a time, and I'll have one team at a time "run the gauntlet" against the remaining teams that have not played out their full schedules.
So the first team scheduled to run the gauntlet is team A. They win their first match against team F, and then draw against all other remaining teams (as those teams will have entered their match with A on 0 points). Team A finishes with a 1-0-4 record for 7 points.
B runs the gauntlet next, starting with a draw against F (1 point vs. 0 at the time of their matchup) to get to 2 points. The other teams left on their schedule (C, D, E) all have 1 point, so all these matches can end in a draw as well. B goes a "perfect" 0-0-5 for 5 points. At this point, 8 out of the 9 matches played have ended in draws.
From here, the pattern is clear: don't let F run the gauntlet, but schedule any other team to do it. Have the scheduled team play and draw against F first, as F will always have 1 point less than the scheduled team at the time of their matchup. The scheduled team, now having a lead over all remaining teams on their schedule after the match against F, can draw all their remaining matches.
Rinse and repeat until the end, and we see that 14 out of the 15 matches can be drawn. Maximum draws = 14.
And as for the final standings... team A wins with 7 points, B-E tie for 2nd with 5 points, and team F is last with 4 points.
Hi, thanks for the comment. But football tournaments do not take place with games in that order. Usually there is a first round of 3 games involving all 6 teams - the teams are shuffled and play round 2, and so on until round 5. Like the World Cup or even the Premier League.
This question was adapted from, if I recall, an Argentinian mathematics competition, and the original question also assumes that people know how such tournaments take place. Perhaps because it is a footballing nation.
Football = soccer
:-)
Rats! I keep enough tabs on sports that I knew of the simultaneous matches and why they're done that way (one obvious reason would be to make match-fixing for the benefit of outside bettors more difficult). I guess that the soccer powers don't appreciate schedule-fixing either, even if it's in the name of math. :P
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