Clinching early
There are 8 teams in a certain sports league. Each team plays 22 games (11 home, 11 road) against every other team, for a total of 154 games. All games end in either a win or a loss. At all times, the teams have all played the same number of games (all 4 games are played simultaneously on the same day). One year, the Dwarves absolutely dominate, winning every game they play.
What is the minimum number of games into the season (in total) that the Dwarves could have clinched a tie for first--meaning, no one else can better their record even if the Dwarves lose all their remaining games and the other team wins every one of their remaining games?
The answer is 98.
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1 solution
Each other team plays 1 / 7 of their games against the Dwarves and 6 / 7 of their games against teams that are not the Dwarves. We are given that they lose every game against the Dwarves. To make sure that the Dwarves clinch as early as possible, every other team has to win exactly 1 / 2 of their games when playing teams not the Dwarves, so 3 / 7 of their games total. Therefore, the Dwarves gain a 4 game cushion with every 7 games played. Thus, the quickest they can clinch a tie is 9 8 games in.