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In the 40 previous editions only twice has a team trailing by more than two points after the first two days come back to win.
But with late swings on Saturday evening giving the USA a 9½-6½ lead going into the final afternoon, Europe will need to win at least seven and-a-half of the 12 points on offer to hang on to a trophy they have lost only once in the last 14 years.
Rory McIlroy will lead the attempted fight-back in an alluring match-up against Patrick Reed, before Open champion Henrik Stenson takes on Jordan Spieth and Thomas Pieters plays JB Holmes.
A European comeback is not impossible - at Medinah in 2012 they had trailed 10-4 late on the Saturday, and began the singles four down before storming through for a famous victory.
Team talisman McIlroy has led the European charge superbly thus far, winning three points from a possible four, and his foursomes and fourball partner Pieters has also been in the sort of dead-eyed form that belies his rookie status.
Yet with the other two of captain Darren Clarke's wildcards, Martin Kaymer and Lee Westwood, struggling to justify their selection and the US inspired by the stunning performance of the pugnacious Reed, home captain Davis Love will know that his team has a wonderful chance to win only their third Ryder Cup in the last 21 years.
All 12 of Love's players won points in the first two days, while five of Clarke's - Kaymer, Westwood, Danny Willet, Matthew Fitzpatrick and Andy Sullivan - are yet to score.
Clarke, who won four of his five Cups as a player and was a vice-captain at Medinah in 2012, remains defiant despite the disappointment of losing Saturday afternoon's fourballs session 3-1 when at the midway point it had looked as if Europe might have reversed that score.
He said: "We're going to have to work hard and play very, very well, but it's been done before from a worse position - we have a big task ahead of us, but the guys are capable of doing it.
"With the team that's here this week, I don't need to pick them up. Those guys fully believe that they have the talent and the desire to put a result in."
Love was the US captain on that stunning afternoon at Medinah, one of many high-profile US golfers so scarred by the succession of recent defeats that they set up an 11-man task-force to put it right this time.
With only five points required from the 12 on offer on Sunday he is close to achieving that goal and the raucous capacity crowds around this long course are likely to once again offer the most passionate and partisan of support as Clarke puts his big hitters out early.
Love said: "I've obviously been around a lot of teams so it was no secret that they (Europe) were going to load the boat.
"So our problem, was we say, all right, we're going to put the six guys out that are playing really well.
"It's nice to go in on a high, but we know the deal. There are still 12 points to be played, and you can't get off the game plan - you have to keep grinding it out for one more day."
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The magic word at Ryder Cups is momentum, and on Saturday the US snatched that back by winning all three of the last matches on the course having taken only two and-a-half points from the previous nine.
One match in particular that appeared to have cost them was when Sergio Garcia and Rafa Cabrera-Bello came from four down with six to play against Reed and Jordan Spieth to snatch an unlikely half-point in the foursomes and make the overall score at lunchtime 6½-5½.
But two short missed putts from Westwood handed JB Holmes and Ryan Moore both the 17th and 18th holes of their fourballs to hand over the match when the Englishman and partner Willett had not once been behind in the round.
Clarke now needs the early singles matches to go his way to both seize back that impetus and silence a crowd that has celebrated every missed European putt with almost as much fervour as when an American putt has dropped.
With his decision to pick Kaymer and Westwood for the afternoon fourballs also likely to come under scrutiny in the event of a defeat, he also requires a dramatic turnaround from those experienced men as well as an unprecedented display from his six rookies - four of who are out in the last five singles matches.
Europe must scale a mountain if not find another miracle to match that of four years ago if they are to retain the Ryder Cup on Sunday.