Dickens uses the word "conundrum". It occurred to me that I had no idea of the etymology, but sounds a bit like Latin.
American Heritage® Dictionary of the English Language: Fourth Edition. 2000.

conundrum

SYLLABICATION: co·nun·drum
PRONUNCIATION: k-nndrm
NOUN: 1. A riddle in which a fanciful question is answered by a pun. 2. A paradoxical, insoluble, or difficult problem; a dilemma: “the conundrum, thus far unanswered, of achieving full employment without inflation” (Arthur M. Schlesinger, Jr.).
ETYMOLOGY: Origin unknown.

Offhand, I can't remember any conundrums (conunda?) for which answer is a pun. The only one I remember at the moment is "Why is Nancy Reagan like the Panama Canal?"
Answer: "The Panama Canal is a busy ditch, and she's vice versa." But that answer doesn't seem to be a pun.
Got a better example?

Dear Doc

I can't remember specific examples, but Alice in Wonderland is full of conundrums.

Remember the school being run by a turtle? Why? "Because he tortoise" (probably need Brit pronunciation for that). But it's full of that sort of stuff, and I'd guess that the first definition you have may well have come from the Victorian fascination with word games.

cheer

the sunshine warrior

Why is a virgin like a sneeze?
Gesundheit.

And the was the proverbial dumb blonde who insisted that her boyfriend wear a conundrum. Not so dumb, after all.
Incidentally, I despise the merriment directed at blondes.
It has been said Gentlemen Prefer Blondes. I married one, but not because of tint of her chevelure.