Magic number (sports)

Blazers' Playoff Magic Number is Not 4 - Blazer's Edge

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In certain sports, the magic number is the number used to indicate how close the teams are up front to win the division title and/or playoff spot. This is an additional total victory by the front-run team or any additional loss (or combination thereof) by the rival team after which it is mathematically impossible for the rival team to win the title in the remaining number of matches (assuming some very unlikely events such as disqualification or expulsion of competition or retroactive game seizures do not occur). The magic numbers are usually limited to sports where every game results in a win or loss, but not a tie.

Teams other than the front-running team have what's called the elimination number (or "tragic number" ) (often abbreviated E # ). This number represents the number of wins by the main team or the loss by the companion team that will eliminate the counter team. The largest number of eliminations among the non-first team is the magic number for the first team.

The magic number is calculated as G 1 - W _A i> B , where

G is the total number of games in the season
W _A is the number of victories Team A has in the season
L _B is the amount of loss Team B has in the season

For example, in Major League Baseball there are 162 games in one season. Suppose the top of the division standings at the end of the season are as follows:

Then the magic number for Team A to win the division is 162 1 - 96 - 62 = 5.

Any combination of wins by Team A and a loss by Team B for a total of 5 makes it impossible for Team B to win the division title.

"1" in the formula serves to remove the bond; without it, if the magic number decreases to zero and stays there, the two teams in question will end up with identical records. If circumstances dictate that the front-line team will win the tiebreak regardless of future results, then the addition of constant 1 can be eliminated. For example, the NBA uses a complicated formula for disconnecting, using some other achievement statistics in addition to the overall win/loss record; but the first tiebreaker between the two teams is their head-to-head record; if the frontrunning team has achieved a better head-to-head record, then 1 is not required.

The magic number can also be counted as W _B GR _B - W _A 1, where

W _B is the number of wins Team B has had in season
GR _B is the number of games left for Team B in season
W _A is the number of victories owned by Team A in season

This second formula basically says: Assume Team B wins every remaining match. Count how many teams must win to exceed the maximum total of team B of 1 . Using the above example and with the same 162 season, team B has 7 games left.

The magic number for Team A to win the division is still "5": 93 7 - 96 1 = 5.

Team B can win as many as 100 matches. If Team A wins 101, Team B is omitted. The magic number will decrease with Team A's victory and will also decrease with Team B losses, as the total maximum winnings will be reduced by one.

The above variations are seen in the relationship between the losses of both teams. The magic number can be calculated as L _A GR _A - L _B 1, where

L _A is the amount of loss that Team A had in season
GR _A is the number of games left for Team A this season
L _B is the number of losses held by Team B in season

This third formula basically says: Assume Team A loses every game left. Calculate how many B teams need to lose to exceed the maximum of a team A total of 1 . Using the above example and with the same 162 season, Team A has 8 games left.

The magic number for Team A to win the division is still "5": 58 8 - 62 1 = 5. As you can see, the magic number is the same whether counting based on the potential victory of the leader or the potential loss of the trailing team. Indeed, mathematical evidence will show that the three formulas presented here are mathematically equivalent.

Team A could lose as many as 66 matches. If Team B loses 67, Team B is eliminated. Again, the magic number will decrease with Team A's victory and will also decrease with Team B.'s loss.

In some sports, the bond is broken by an additional play-off match between the teams involved. When a team gets to the point where its magic number is 1, it is said to have "won a tie" for a wild division or card. However, if they end the season with another team, and only one qualifies for the playoffs, an additional playoff game will erase that "clinching" for the losing team in the playoff game.

Some sports use a tiebreaker formula instead of doing a one-match playoff. In such a case, it is necessary to look beyond the lost-win record of the team to determine the magic number, since the team that already secured itself the edge in the tiebreak formula does not need to enter "1" in calculating the magic number. For example, assume a basketball league that plays 82 seasons without a one-game tiebreak shows the division standings at the end of the season as follows:

Suppose further that the first step in the league tiebreaker formula is the result in a head-to-head meeting. Team A and Team B have met four times during the season with Team A winning three out of four matches. They are not scheduled to meet again in the regular season. Therefore, Team A holds a tiebreaker advantage over Team B and only needs to finish with the same number of wins as Team B to be placed in front of Team B in the standings. Therefore, we can calculate the magic number of Team A as 82 - 60 - 20 = 2. If Team A wins two of the remaining seven games, it will finish 62-20. If Team B wins all seven games remaining, it will also finish 62-20. However, since Team B lost in a tiebreak on immediate results, Team A was the division winner.

By convention, magic numbers are usually used to describe first-team teams only, relative to the team they lead. However, the same mathematical formula can be applied to any team, team bound to lead, as well as team tracking. In this case, the team that is not in the first position will depend on the main team to lose some games so it can catch up, so the magic number will be greater than the number of games remaining. In the end, for teams that are no longer competing, their magic count will be greater than their remaining game remaining for first-team teams - which is impossible to overcome.

Video Magic number (sports)

Derivation

The formula for magic numbers is derived as follows. As before, at some point in season let Team A have W _A win and L _{< i> A} loss. Suppose that at a later time, Team A has additional w _A and l _A additional loss, and defines the same W _B, L _{< i> B}, w _B, l _B for Team B. The total number of wins Team B needs to perform by ( W _A w _A) - ( W _B w _B). Team A wins when this number exceeds the number of remaining matches of Team B, because at that point Team B can not make a deficit even if Team A fails to win more games. If there is a total of G games in this season, then the number of games left for Team B is given by G - ( W _{B w} _B L _{B >} l _B). Thus the conditions for Team A to reach are ( W _i) - ( W _B w _{B >}) = 1 G - ( W w < sub> L _B l _B). Canceling the generic term, we got w l _B = G 1 - W _A - L _B, which sets the magic formula formula.

Maps Magic number (sports)

Played Quirk Game
In the following example, Team A's Magic Number is 5, because although it can eliminate Team B in second place in 4 additional games, it takes 5 games to ensure eliminate the third position of Team C. Counting the magical number requires using the lowest amount of loss among other competing teams : 162 1 - 88 - 70 = 5.
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Subtlety

Sometimes a team can show up to have a mathematical chance to win even though they have actually been eliminated, due to scheduling. In this Premiership Baseball scenario, there are three matches left in this season. Teams A, B and C are assumed to be entitled only to division championships; teams with better records in other divisions have earned two available wild card spots:

If Team C wins three matches remaining, it will end at 88-74, and if both Team A and B lose three matches left, they will finish at 87-75, which will make Team C the division winner. However, if Team A and B play against each other over the last weekend (in the 3-game series), it is impossible for both teams to lose the three remaining matches. One of them will win at least two games and thus earn a division title with a 90-72 or 89-73 record. The more direct consequence of this situation is that it is also impossible for Team A and B to finish the game with each other, and Team C can not win the division.

One can say with certainty whether a team has been eliminated by using algorithms for maximum flow problems.

The addition of the second Wild Card team made upside-down scenarios (where a team really had grabbed the post-positioning venue even though it appeared they could still be omitted) maybe in baseball. In this scenario for Wild Card:

If Team B and C play their last three games against each other and all the other teams have reached their division or mathematically have been eliminated from the capture of Team A, then Team A will grab at least the second Wild Card position because it is unlikely to happen. Teams B and C both won enough games to capture Team A.

The reverse scenario is more common in sports with post-passenger berths, favoring teams in final playoff positions but being chased by teams still playing with each other. Sometimes, both scenarios can occur simultaneously. In the following National Basketball Association scenario for teams placed in the seventh to the tenth position in the conference standings:

If Team B and C have to play one of their last two matches against each other and Team A holds a tiebreak against Team B, C and D, then Team A will take the playoff spot because they can not be beaten by both B and C teams. Also, if Team D does not hold a tiebreak against one of Team A, B and C then it will come out of a playoff dispute as it can not take over both Team B and C.

Similar scenarios sometimes occur in European football leagues and other competitions that use promotion and degradation. In this scenario for a 20th team soccer league that plays a double-round robin format, award three points for winning and one for a draw and throw a team of ranks 18, 19 and 20:

If Team A loses in the last two games, it will finish with 38 points while if Tim D wins the last two games, it will end with 34. However, regardless of the goal difference or the other tiebreak, if Team B and C still have to play one Team A is safe from relegation because Team B and C can not both reach 38 points while Team D will be relegated as Team B and C can not finish both with less than 35 points.