How was Sauron's plan foiled
 

Was Sauron's plan with the rings foiled because of the 3 elven rings? Were these the rings that detected him putting on the one ring? Sauron didn't know about these did he, and if there was a chance the other 16 could detect then he would have been more careful? Though when Frodo put the One ring the Nazgul could find him? So it could have been one of the others?