Magic number (sports)

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In certain sports, a magic number is a number used to indicate how close a front-running team is to clinching a division title and/or a playoff spot. It represents the total of additional wins by the front-running team or additional losses (or any combination thereof) by the rival teams after which it is mathematically impossible for the rival teams to capture the title in the remaining number of games, assuming some highly unlikely occurrence such as disqualification or expulsion from the competition or retroactive forfeiture of games does not occur.

The widespread use of magic numbers is generally limited to sports where games only count in the standings when the result is a win and a loss. Magic numbers are not usually used in sports where teams can be credited in some manner for part-wins in case of results such as ties and overtime losses. It could also be referred to as the "clinching number".

Teams other than the front-running team have what is called an elimination number (or "tragic number") (often abbreviated E#). This number represents the number of wins by the leading team or losses by the trailing team which will eliminate the trailing team. The largest elimination number among the non-first place teams is the magic number for the leading team.

The magic number is calculated as G + 1 − W_A − L_B, where

• G is the total number of games in the season
• W_A is the number of wins that Team A has in the season
• L_B is the number of losses that Team B has in the season

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

Then the magic number for Team B to be eliminated is 162 + 1 − 96 − 62 = 5.

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

The "+1" in the formula serves the purpose of eliminating ties; without it, if the magic number were to decrease to zero and stay there, the two teams in question would wind up with identical records. If circumstances dictate that the front-running team would win the tiebreaker regardless of any future results, then the additional constant 1 can be eliminated. For example, the NBA uses complicated formulae for breaking ties, using several other statistics of merit besides overall win–loss record; however the first tiebreaker between two teams is their head-to-head record; if the front-running team has already clinched the better head-to-head record, then the +1 is unnecessary. In 2022, Major League Baseball introduced tiebreaking scenarios (such as head-to-head for division ties) that made the use of the "+1" pointless (as Game 163 was eliminated).

The magic number can also be calculated as W_B + GR_B − W_A + 1, where

• W_B is the number of wins that Team B has in the season
• GR_B is the number of games remaining for Team B in the season
• W_A is the number of wins that Team A has in the season

This second formula basically says: Assume Team B wins every remaining game. Calculate how many games team A needs to win to surpass team B's maximum total by 1. Using the example above and with the same 162-game season, team B has 7 games remaining.

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 games. If Team A wins 101, Team B is eliminated. The magic number would decrease with a Team A win and would also decrease with a Team B loss, as its maximum win total would decrease by one.

A variation of the above looks at the relation between the losses of the two teams. The magic number can be calculated as L_A + GR_A − L_B + 1, where

• L_A is the number of losses that Team A has in the season
• GR_A is the number of games remaining for Team A in the season
• L_B is the number of losses that Team B has in the season

This third formula basically says: Assume Team A loses every remaining game. Calculate how many games team B needs to lose to surpass team A's maximum total by 1. Using the example above and with the same 162-game season, team A has 8 games remaining.

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 calculating it based on potential wins of the leader or potential losses of the trailing team. Indeed, mathematical proofs will show that the three formulas presented here are mathematically equivalent.

Team A can lose as many as 66 games. If Team B loses 67, Team B is eliminated. Once again, the magic number would decrease with a Team A win and would also decrease with a Team B loss.

In some sports, ties are broken by an additional one-game playoff(s) between the teams involved. When a team gets to the point where its magic number is 1, it is said to have "clinched a tie" for the division or the wild card. However, if they end the season tied with another team, and only one is eligible for the playoffs, the extra playoff game will erase that "clinching" for the team that loses the playoff game.

Some sports use a tiebreaker formula instead of staging a one-game playoff. In such cases, it is necessary to look beyond the won-lost records of the teams to determine the magic number, since a team that has already guaranteed itself the edge in the tiebreaker formula would not need to include "+1" in calculating its magic number. For example, assume a basketball league that plays an 82-game season with no one-game tiebreakers shows division standings late in the season as follows:

Suppose further that the first step in the league's tiebreaker formula is results in head-to-head meetings. Team A and Team B have met four times during the season with Team A winning three of the four games. They are not scheduled to meet again in the regular season. Therefore, Team A holds a tiebreaker edge over Team B and only needs to finish with the same number of wins as Team B in order to be placed ahead of Team B in the standings. Therefore, we can calculate Team A's magic number as 82 – 60 – 20 = 2. If Team A wins two of its seven remaining games, it will finish 62–20. If Team B wins all seven of its remaining games, it will also finish 62–20. However, since Team B loses the tiebreaker on head-to-head results, Team A is the division winner. In cases where the winners of potential tiebreakers have not yet been determined (for example, because the teams still have some games to play against each other) the usual convention is to calculate the magic numbers of the teams involved as if they will lose the tiebreaker, and to calculate the elimination numbers of such teams as if they will win the tiebreaker.

By convention, the magic number typically is used to describe the first place team only, relative to the teams it leads. However, the same mathematical formulas could be applied to any team, teams that are tied for the lead, as well as teams that trail. In these cases, a team that is not in first place will depend on the leading team to lose some games so that it may catch up, so the magic number will be larger than the number of games remaining. Ultimately, for teams that are no longer in contention, their magic number would be larger than their remaining games + the remaining games for the first place team — which would be impossible to overcome.