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You're both wrong. The player with the ball farthest from the hole plays first.
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I agreed with this one except for when it is reasonable that you can make the shot and there is a ball in the way. In this case the close one should finish out.
Although, it seems I am wrong. The Pro Putters Assoc. has a rule book online. And it states that one player plays the entire hole and then the next player does.
http://www.proputters.com/downloads/..._Book_2003.pdf
Go figure.