Does your brain hurt trying to work out the permutations for the ATP Finals?

Roger Federer beat Dominic Thiem on Tuesday to keep alive his hopes of making it through to the semi-finals at the 02 Arena in London.

But whether or not he will qualify is horribly complicated.

You can’t say he is through if he beats Kevin Anderson in his final group match on Thursday, because there are several different scenarios that can happen.

Thiem faces Kei Nishikori in the other Group Lleyton Hewitt match on Thursday and either of those could progress.

Federer could win in straight sets and still not make it through. He could lose in straight sets and qualify. So how does that work?

Before we look at that, here’s how people reacted on social media after the ATP tweeted this graphic:

‘My brain hurts’

Some of you just replied with emojis of confused faces.

But many of you responded with funny gifs including this one from Abel Morton.

And this one from Roger Buchana.

Though most of you just said what we were all thinking…

Optimist: I have a headache!

Marcou: I’m lost!

Adeyanju Michael: This is Maths 🙁

Aditya Jain: Don’t understand anything.

N Kessler: My brain hurts.

So, is there an easy way to understand it?

In short, no.

Here’s our attempt at a ‘simple‘ breakdown of how the top two are decided in each of the two four-player groups:

• Greatest number of wins
• If two players are tied: Head-to-head results
• If three players are tied: 1) Highest percentage of sets won; 2) Highest percentage of games won; 3) Players’ ATP Rankings coming into the tournament

So with that in mind, Federer will go through if any of these situations happen:

1. He wins in two sets and Thiem beats Kei Nishikori in two.
2. He wins in two sets and Thiem wins in three.
3. He wins in two sets and Nishikori beats Thiem in two sets. Highest % of games won will then be the decider.
4. He wins in two sets and Nishikori wins in three.
5. He wins in three sets and Thiem wins in two.
6. He wins in three sets and Thiem wins in three.

This is how the group currently stands:

If you’re still not sure what’s going on, the best thing to do is to follow our guide on the group standings, results and fixtures and just trust us when we say Federer is through or not.

Roger Federer recovered from his first-match nightmare to keep alive his hopes at the ATP Finals with a comfortable victory over Dominic Thiem.

Switzerland’s Federer, knowing defeat would mean he would not advance from his group, beat the Austrian 6-2 6-3.

The six-time champion had slumped to a shock defeat by Kei Nishikori on Sunday but he cut down on his errors to win.

Next up for Federer on Thursday is Kevin Anderson, who beat Nishikori 6-0 6-1 earlier in the day.

The action in London continues in the other group on Wednesday, when world number one Novak Djokovic plays Alexander Zverev, which you can watch on BBC Two (14:00 GMT) and follow with live text commentary, before Marin Cilic takes on John Isner with BBC Radio 5 live sports extra commentary online (20:00).

• Is your brain hurting with ATP Finals permutations?
• Anderson thrashes Nishikori at ATP Finals
• Live scores, schedule and results
• Alerts: Get tennis news sent to your phone

Federer rediscovers positive mindset

Just before walking on to the court, Federer spoke of the need to be “more positive” in his mindset.

He arrived with a little smile and a few waves before making his intentions clear from the very first game where he carved out a break point.

Although he failed to convert that one, he made no mistake in Thiem’s next service game and celebrated with a fist pump when the Austrian sent a forehand long.

This was a completely different Federer to the one who had looked frustrated and grouchy against Nishikori in his opening round-robin match on Sunday.

And it was not just his mood that had improved considerably; his statistics had too.

Gone were the 34 unforced errors – instead a much more disciplined 11 – and with that a superb rate of 86% of first-serve points won and 81% of second-serve points.

He looked more like his old self and was helped by Thiem’s errors, summed up when the Swiss sealed victory after the world number eight hit a forehand long.

“It feels good. I am very happy that I showed a reaction after the last match,” Federer told Sky Sports.

“Today I was more positive and happy on the court. I love playing in London and I had to remind myself what a privilege it is. I hope I showed it.”

Next up Anderson – and some maths

Had Federer lost to Thiem, he would have failed to reach the semi-finals at the season-ending event for just the second time in his record 16 appearances.

And while victory keeps him in the hunt in the Lleyton Hewitt group, his path to the last four is still far from certain.

With one match to play, all four players in the group can still mathematically qualify – and it has the potential to get very complicated.

It is possible that Federer could beat Anderson but still not advance if Nishikori beats Thiem, depending on who wins how many sets and games.

There is even a scenario whereby Federer can lose to Anderson in three sets but go through if Thiem beats Nishikori in two.

The meeting between 20-time Grand Slam champion Federer and world number six Anderson is a re-run of this year’s Wimbledon quarter-final, where the South African fought back from two sets down to record a famous victory.

And after his ruthless showing against Nishikori, unbeaten Anderson is a tricky prospect.

“I hope it is not a match like Kevin Anderson played this afternoon,” Federer said. “It is never easy to play against him, especially indoors. I am happy I am getting a chance to play him again and have a re-match [from Wimbledon].”

How the top two are decided in each group:

• Greatest number of wins
• Greatest number of matches played (eg. a 2-1 won-loss record beats a 2-0 won-loss record; a 1-2 record beats a 1-0 record).
• If two players are tied: Head-to-head results
• If three players are tied:
• – 1) Highest percentage of sets won
• – 2) Highest percentage of games won
• – 3) Player’s ATP Rankings coming into the tournament

Analysis

BBC tennis correspondent Russell Fuller

This was another lacklustre performance by Thiem, but Federer looked slick and sharp: in short, a very different player to the one beaten by Kei Nishikori on Sunday evening.

Federer gave up very few points on serve and broke Thiem four times, and yet still has plenty of work to do to reach the semi-finals.

He has the unbeaten Kevin Anderson to worry about, as well as some advanced arithmetic.

In theory, Federer could win in straight sets on Thursday and still go out, but he could also lose in straight sets and still go through.

Britain’s Jamie Murray reached the doubles semi-finals of the ATP Finals in London for a third successive year.

The Scot and Brazilian partner Bruno Soares beat Colombia’s Juan-Sebastian Cabal and Robert Farah 6-4 6-3.

Breaks in the 10th game of the first set and fourth game of the second proved decisive as the fourth seeds sealed victory in one hour 13 minutes.

Murray served out the match, sealing victory when Cabal sent a forehand into the net.

The pair had also won a tight opening match against Raven Klaasen and Michael Venus on Sunday, meaning they make sure of progressing with a match to spare.

• Anderson thrashes Nishikori at ATP Finals
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Analysis

Mike Dickson of the Mail on BBC Radio 5 live sports extra

It was a really excellent performance. They weathered that storm in the first set when the Colombians came back at them and won it at the end. It was then plain sailing in the second set.

You have to think, is this going to be their year? You could not have asked for more than that in this game.