The phrase "famous conundrum" can be replaced with a variety of synonyms, depending on the context. A conundrum is essentially a difficult problem or puzzle, so synonyms that might fit include enigma, puzzle, riddle, mystery, or paradox. If the conundrum is particularly well-known, it might be described as an "iconic puzzle" or a "legendary mystery." In some cases, synonyms that focus on the confusion or ambiguity at the heart of the conundrum might be more appropriate, such as "mind-boggler," "brain-teaser," or "head-scratcher." Ultimately, the choice of synonym will depend on the specific nature of the conundrum being discussed.