1. a puzzle or problem with a difficult or impossible solution.
Who or what produced these mysterious crop circles is still a conundrum for scientists.
How to continue to provide necessary services to the people without raising taxes has always been a conundrum for the city government.
2. a riddle whose answer contains or is a pun.
Here’s a conundrum: What is the difference between a jeweler and a jailer? The answer is that a jeweler sells watches while a jailer watches cells.