[Computational Complexity] Magic Numbers Revisited
Google for Yankees Magic Number and short down the list you'll find my 2004 post The Beauty of the Magic Number.
As I write this the Yankees have 88 wins and Boston with 68 losses which means the Yankees current magic number is 162+1-(88-68)=7. This means any combination of 7 Yankees wins and Red Sox losses would guarantee that the Yankees win their division.
Yesterday I got the following email.
I'm in disagreement with my brother—and, apparently, all major media—regarding the Yankees' magic number. My reasoning is that if the Red Sox win all their remaining games, that's 94 victories. Since the Yankees lead the season series and will thus win the division in the case of a tie with Boston, they'll clinch the division with 94 wins. They have 88, so only need eight more…thus, a magic number of six.The standard magic number is for winning the division outright. According to Wikipedia
The formula you reference (with the "+1"), which I guess others use as well, doesn't account for the fact that a team only need to tie for first if they've won the season series vs. the other team.
In the event of a tie in the standings at the close of the regular season, league rules provide for a one-game playoff (with the home field determined by coin flip) to determine which of two teams participate in the Division Series. If three teams are involved in a tie, a two-game playoff may be played. If two teams are tied, but a tiebreaker would result in both participating in the Division Series anyway (due to one being division champion and the other being wild card), then no playoff is played and seedings are determined by head-to-head record.Since the Red Sox are unlikely to get the wild card this year, the Yankees would have to win a playoff game in case of a tie so they would need the extra win anyway and the magic number of 7 is valid.
The situation is different in the AL Central where the second place team will likely win the wild card. Detroit has 87 wins and Minnesota has 60 losses and Detroit has the better head-to-head. So Detroit only needs 162-(87+60)=15 wins and/or Twins losses to clinch the division since they win the division in case of a tie. No "+1" here.
Now if the Twins fade and my White Sox (with 62 losses) get into contention then Detroit needs 162+1-(87+62)=14 wins and/or White Sox losses to clinch over the White Sox. Why? Because the White Sox will have the better head-to-head over Detroit.
Nothing like Major League Baseball and their wild-card rules to ruin the beauty of the magic number.
Posted by Lance to Computational Complexity at 9/14/2006 08:13:00 AM