| has abstract
| - 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). Magic numbers are generally confined to sports where each game results in a win or a loss, but not a tie. 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 − WA − LB, where
* G is the total number of games in the season
* WA is the number of wins that Team A has in the season
* LB 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 WB + GRB - WA + 1, where
* WB is the number of wins that Team B has in the season
* GRB is the number of games remaining for Team B in the season
* WA 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 LA + GRA - LB + 1, where
* LA is the number of losses that Team A has in the season
* GRA is the number of games remaining for Team A in the season
* LB 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. 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. (en)
- Dans certains sports, un chiffre magique, ou nombre magique, est un nombre utilisé pour illustrer à quel point un club est près de s'assurer d'une première place ou d'une qualification aux séries éliminatoires. (fr)
- ( 다른 뜻에 대해서는 마법수 문서를 참고하십시오.)( 컴퓨터 과학 용어에 대해서는 매직 넘버 (프로그래밍) 문서를 참고하십시오.)
매직 넘버(magic number)란 스포츠 리그에서 우승하거나, 다른팀을 앞서거나 포스트시즌 진출이나 승격, 잔류등 특정 자격에 대하여 앞으로 최소 몇경기만 이기면 어느팀을 상대로 이기든지 목표를 달성할 수 있는지를 나타내는 숫자이다. 반대로 최소 몇경기를 지면 다른팀의 경기에 상관없이 목표를 얻지 못하는 것은 라고 한다. (ko)
- マジックナンバー（magic number）とは、プロ野球の用語で、「他のチームの試合結果に関わらず、自チームの優勝に繋がる勝利数」と言える勝ち数を意味する。 日本では一般にマジックと呼ばれる。 他チームが残り試合に全勝することを仮定して算出される数であるため、実際に優勝までにあと何回勝たねばならないかよりも大きめの値となるのが普通である。 日本では他の全チームに自力優勝の可能性がなくなった状況でのみこの値を用い、この条件を満たすことをマジックナンバーが「点灯」したという。野球チームは優勝までに、マジック点灯⇒マジックナンバーを減らす⇒優勝決定という経過を通常たどる。マジックナンバーはチームが優勝するまでの道筋として用いられる。 アメリカ合衆国では、自力優勝の条件が満たされない場合でもマジックナンバーを用いるために、そもそも「マジック点灯」と言う概念はなく、野球以外のスポーツでもマジックナンバーを用いる。 (ja)
- Em certos esportes, um número mágico (magic number) é um número indicado para indicar quão perto um time na dianteira está de conquistar um título na temporada. Ele representa o total de vitórias adicionais pelo líder ou derrotas adicionais pelo time adversário que depois tornam matematicamente impossível para o adversário capturar o título nos jogos restantes. Isto assume que cada jogo resulta em uma vitória ou derrota, mas não em empate. Os outros times que não o líder têm o que é chamado um número de eliminação/trágico (elimination/tragic number) (por vezes abreviado "E#"). Este número representa o número de vitórias pelo líder ou derrotas pelo time perseguidor que eliminará o perseguidor. O número de eliminação do time em segundo lugar é exatamente o número mágico do time em primeiro. O número mágico é calculado como G + 1 − WA − LB, onde:
* G é o número total de jogos na temporada;
* WA é o número de vitórias que o time A tem na temporada;
* LB é o número de derrotas que o time B tem na temporada. Por exemplo, na Major League Baseball há 162 jogos numa temporada. Suponha que o topo da classificação da divisão mais tarde na temporada seja assim: Então o número mágico para o time "A" ganhar a divisão é 162 + 1 − 96 − 62 = 5. Qualquer combinação de vitórias pelo time "A" e derrotas pelo time "B" totalizando 5 fará com que seja impossível ao time "B" ganhar o título da divisão. O "+1" na fórmula serve o objetivo de eliminar empates; sem ele, se o número mágico mágico caísse a zero e permanecesse assim, os dois times em questão acabariam com marcas idênticas. Se as circunstâncias ditam que o líder ganharia o desempate apesar de qualquer resultado futuro, então o 1 constante adicional pode ser descartado. Por exemplo, a NBA usa uma fórmula complicada para quebrar empates, usando várias outras estatísticas de mérito além do registro total de vitórias/derrotas; contudo, o primeiro desempate entre dois times é o seu confronto direto; se o time da frente já tem essa vantagem, então o +1 é desnecessário. Em outros esportes, como na MLB, os empates são normalmente quebrados por um extra entre as equipes envolvidas; o +1 adicional não pode ser removido da fórmula a menos que ambos os times sejam da mesma divisão e ambos estejam garantidos nos playoffs (um como vencedor da divisão e outro como repescado). Quando um time chega ao ponto onde seu número mágico é 1, se diz que ele "alcançou um empate" pela divisão ou repescagem. Porém, se eles terminarem a temporada empatados com outro time, e somente um tiver direito aos playoffs, o jogo decisivo extra apagará aquele "alcance" para o time que perde o jogo decisivo. (pt)
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