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Four bowling teams (named A, B, C, and D) will be participating in a round robin (every team plays every other team once, no team is eliminated) playoff. A playoff is called a roll-off in bowling. Each team will play three games. For each game, two teams will bowl each other on lanes 3 and 4, while the other two teams bowl each other on lanes 5 and 6. One of the organizers draws up a complete schedule, showing which teams play which teams, and who bowls on which alley:
| lanes 3 and 4 | lanes 5 and 6 | |
| game #1 | A vs. B | C vs. D |
| game #2 | A vs. C | B vs. D |
| game #3 | A vs. D | B vs. C |
The coach of one team then complains that one of the teams (team A) will get to bowl on the same pair of lanes all three games. In general, repeatedly bowling on the same pair of lanes should be an advantage. Using only the two pairs of lanes mentioned, can you suggest a schedule which will have no team bowling on the same pair of lanes for all three games?
Answer:
This question has come up a couple of times during my bowling life. I have been in several leagues which have schedules divided up into quarters. The winner of each quarter then bowls in a roll-off to determine the champs for the entire year. And if a team wins one quarter, they are generally ineligible to win later quarters. It turns out that the answer to the above question is "No, one of the teams must bowl all three games on one pair of lanes." A proof of this is that after the first game, two teams must move to the opposite pair while two teams must stay put (not move). After the second game, the two teams that stayed put must bowl each other (they haven't bowled each other yet), and one of them must stay put again. By the way, all possible 4-team round robin schedules are equivalent to the schedule that I showed above, with the team names and lane numbers scrambled somehow.