Martin Kaymer won the 2010 PGA Championship but lasted in the top spot only eight weeks.
Westwood has been paired with Martin Kaymer against Phil Mickelson and Dustin Johnson in the first fourball.
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Mickelson, Woods and USPGA champion Martin Kaymer all have a chance to take Westwood's number one spot this week.
Westwood is well placed to retain his ranking, with rivals Martin Kaymer and Phil Mickelson five shots behind him.
Former world No. 1 Martin Kaymer, who uses a traditional putter, also wants to see the issue put to bed.
The US Open winner needs a top-three finish to have any chance of passing Race to Dubai leader Martin Kaymer.
Graeme McDowell (69) is 10 shots back and conceded defeat to Martin Kaymer in the battle to be Europe's number one.
Lee Westwood, Martin Kaymer and Luke Donald took turns before this week.
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Italian brothers Edoardo and Francesco Molinari formed another pair, while world number four Lee Westwood teamed up with USPGA champion Martin Kaymer.
Both Martin Kaymer and Rory McIlroy the most noteworthy and recent examples.
Wild card pick Nicolas Colsaerts of Belgium shot 65 to reach five under, while Germany's Martin Kaymer is a shot further back.
Non-Ameicans Lee Westwood, Martin Kaymer, Luke Donald and Rory McIlroy have all taken turns at the top of the World Golf Rankings.
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Defending champion Rock finished on 76, while Ernie Els, playing alongside Rock and Rose, had a 71, the same as Martin Kaymer.
John Parry led the field by two after a 71, with Ryder Cup man Martin Kaymer firing a 68 to share second spot.
World number three and defending champion Martin Kaymer was the biggest casualty, missing out by a single shot after rounds of 72 and 73.
Defending champion Matteo Manassero of Italy and seventh-ranked Martin Kaymer of Germany both shot two under par rounds of 70 to stay in contention.
Martin Kaymer, who carded a 68 at Kingsbarns, is the highest-placed Ryder Cup player, while England's Ross Fisher carded a 69, also at Kingsbarns.
Ian Poulter and Ross Fisher are tied with Tiger Woods and Steve Stricker, while Lee Westwood and Martin Kaymer lead Phil Mickelson and Dustin Johnson.
McIlroy was second on 16 under, eight shots behind winner Martin Kaymer, thanks to a closing round of 69 which included five birdies and two bogeys.
World No. 1 Martin Kaymer was tied for fifth on nine under with three holes to play, having shot six birdies in an as-yet unblemished scorecard.
Casey, who eight days ago lost his lead on the European "Race to Dubai" money list to German Martin Kaymer, lies 18th in the FedEx Cup standings.
Other players including Martin Kaymer, Phil Mickelson and Steve Stricker had chances to grab the No. 1 ranking this year, but none of them could close the deal.
Meanwhile, Germany's Martin Kaymer has been ruled out of action for at least six weeks after sustaining a foot injury when go-karting in America which will require surgery.
Germany's Martin Kaymer shot a eight-under par 64 to move up into a tie for seventh with South Africa's Charl Schwartzel (69) and America's Charles Howell III (67).
The European team also included Lee Westwood, Rory McIlroy, Martin Kaymer, Ian Poulter, Ross Fisher, Francesco Molinari, Edoardo Molinari, Peter Hanson, Luke Donald, Padraig Harrington and Miguel Angel Jimenez.
Westwood is not the only player in the running to overtake Woods at the top of the world rankings as Germany's Martin Kaymer could also move ahead of the American.
McIlroy, who carded a disastrous 80 in the second round, again showed his mettle to delight the galleries with a four-under 68, but Germany's Martin Kaymer, third overnight, slipped to a 74.
Paul Casey, still battling the rib injury which sidelined him for three months, Volvo World Match Play Championship winner Ross Fisher and Martin Kaymer, who is still struggling with a foot injury, sit on four-under.
Germany's Martin Kaymer shot 68 to finish third on eight under with England's Lee Westwood (71), Sweden's Henrik Stenson (67) and Spain's Alejandro Canizares (71) eight shots back and American Dustin Johnson (69) nine behind.
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