The quick answer, in case you don’t want to read further, is yes, he can, but it’ll be tough.

Andy Murray has laid the gauntlet down.  He has made it challenging for Federer to pass him.  Andy Murray has a 710 point lead heading into the year-end championship in London.  Neither are in the Davis Cup final so there are no other points to be gained.

Andy Murray could have made it impossible for Federer to finish number 3 had he beaten Federer in the semis of Paris.  The gap would be too big to surmount.  Instead, Murray lost to Berdych in the quarters.  Despite this loss, Murray picked up 180 valuable points to increase the distance between him and the Swiss by nearly 200 points and that is key for Murray to end the year ranked number 3.

For Federer finish ahead of Murray, he must earn 720 points more than Murray earns in London.

In London, every round robin match win is 200 points.  This is the cheapest way to pick up 200 points ever.  Beat one guy, and you get nearly the amount of an ATP 250 event.  It allows the top 8 to create distance from their other competitors.  Each win Murray makes in round robin makes Roger’s challenge tougher.

If you reach the final of London, you get 400 points more.  Win the final, and you get 500 points more.  If you go undefeated, you max out with 1500 points (600 from round robin, 400 from getting to the final, 500 from winning).  This is what Federer did last year.  Maxing out is 3/4 of a Slam win, so it’s considerable.  Federer knew this which is why he pushed so hard to do well at London.  Unfortunately, there was Djokovic, otherwise, had Federer returned to his Slam winning ways, he might have regained number 1.

Let’s look at the scenario between Federer and Murray more closely.  If Murray wins twice in round robin, he’ll pick up 400 points  That means Federer needs 1110 points to get to number 3.  Even if Federer sweeps the round robin (600 pts) and reaches the final (400 points), he’ll still be over 100 points short.  Federer has to either hope Murray does awful in round robin play (wins 200 points or less) and even then, he must still reach the final.  Even if Murray doesn’t play at all (o points), Federer must reach the final to overtake Murray.

If Murray reaches the final, then Federer has to get the full 1500 pts to beat Murray.  That’s assuming Murray only gets one win in round robin and the tiebreaks work his way so he qualifies for the semis.  Murray makes Federer’s life tough by winning two matches in round robin play which is how many he won last year.

So can it be done?  Yes.  But basically Federer needs to win London to give himself a good shot at ending the year at number 3.