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According to the FAQ published by both the USGA and the R&A, under Rule 14-1:
Using Club other than Putter on Putting Green?
Q. Is it permissible to use a club other than a putter to play a stroke on the putting green?
A. Yes, the Rules of Golf do not stipulate the type of club that must be used for making a stroke on the putting green or anyplace else on the golf course.
Interesting thread. Many moons ago my boss conned me into taking up golf, a year or two later having sensed my discouragment at not being able to even come close to matching his scores he offered to play a round using just one club against my full bag. Seemed a no brainer to me, we bet a case of bud and we played our home course, guess who won. 18 years later I've yet to beat him, ahh it's a screwey game anyway. Just gotta keep practicing I guess.