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This article is mainly motivated by the urge to answer two kinds of questions regarding the Bundesliga, which is Germany's primary football (soccer) division having the highest average stadium attendance worldwide: "At any point in the season, what is the lowest final rank a certain team can achieve?" and "At any point in the season, what is the highest final rank a certain team can achieve?". Although we focus on the Bundesliga in particular, the integer programming formulations we introduce to answer these questions can easily be adapted to a variety of other league systems and tournaments.

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The original version of the article contained an error in the equation (6b). The reference to the equation was inadvertently given as (5a) instead of (5b). The correct equation is given below: The original article has now been corrected. Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. (6b) min n − ∑ m∈T∖{t} c m s.t. (1), (4), (5b).