How Toni Kroos bent World Cup Group F out of shape and wide open for Germany, Mexico, Sweden and South Korea

If not for his stunning late intervention in Sochi on Saturday night, the picture would be relatively clear

Abacuses, calculators and diplomas in applied mathematics at the ready. If you found yourself confused by Group B’s various permutations in the final minutes of Tuesday night’s matches, you might want to keep a packet of aspirin close to hand for Group F’s shakedown on Thursday afternoon.

Blame Toni Kroos. If not for his stunning late intervention in Sochi on Saturday night, the picture would be relatively clear. Mexico would have already qualified, South Korea would be planning their journey home and Sweden would have the upper hand on the world champions, Germany, who would be staring down the barrel.

Forget that, though. Kroos changed everything. The German’s stoppage-time free-kick was not only spectacular because it was scored at a narrow angle, or because of its wicked whip and trajectory, or even its late arrival in the fifth of five minutes added on. It was also spectacular because it altered the immediate outlook for every member of Group F.

On Thursday, Mexico play Sweden in Ekaterinburg while Germany play South Korea in Kazan. None of the four have qualified, none of the four have been eliminated, all retain at least some hope of reaching the round of 16, but the path is more straightforward for some than it is for others.

Mexico need a point to progress as group winners, though a defeat will be enough to qualify if Germany fail to win. Sweden will progress if they win or better Germany’s result, and will top the group if they beat Mexico and Germany fail to win.

Toni Kroos's late goal against Sweden changed the group (Getty)

Germany, meanwhile, need to better Sweden’s result or win by two goals or more to qualify. South Korea can still progress despite losing their opening two games, amazingly, but need to beat Germany and a Mexico victory against Sweden.

There is also the possibility that both Germany and Sweden draw. In that case, the team in the highest-scoring draw progresses with Mexico. If both draw by the same scoreline, Germany’s head-to-head record against Sweden would see them progress.

That, unfortunately, is the easy part. The headaches come with certain combinations of scorelines that leave two or more teams with identical records. If, for example, both Sweden and Germany win 1-0, there will be a complete three-way tie at the top of the group with Mexico. The three teams would enter a mini-league, from which Germany and Sweden would qualify on goals scored between the three.

Or, consider if Sweden and Germany win 2-1. Again, there would be a three-way tie at the top. Sweden would win the group on goals scored between the three. Germany and Mexico, however, would be tied on two goals each and second place would be decided on fair play points. As it stands, Germany – who had Jerome Boateng sent off late on against Sweden – would be eliminated.

A three-way tie for second is not beyond the realms of possibility either. If South Korea beat Germany and Sweden lose, all three will be level on three points. Korea, in this case, would need to win by two goals or more to guarantee qualification. A 1-0 win for Korea and a 1-0 Sweden victory would leave fair play points to decide Sweden and Germany’s fate.

Jerome Boateng was sent off against Sweden (Getty Images) (FIFA via Getty Images)

Unsurprisingly, the various permutations – and alas, there are more than those noted here – were not entirely clear in the immediate aftermath of Kroos’s free-kick. It required a Fifa press release on Monday morning to clarify the situation once and for all. As the release notes, there are three other scenarios that would produce a ‘perfect tie’ in Group F, which would leave qualification decided by a drawing of lots.

Ultimately, it is perhaps easiest to say this: Mexico, Group F’s most impressive side so far, should be confident of beating Sweden, and Germany, having survived their scare, will expect to beat South Korea. In that scenario, both Mexico and Germany progress. Yet if you have learned anything from this preview, it should be that the World Cup is rarely that simple.