conundrum

American Heritage® Dictionary of the English Language, Fourth Edition

1. n. A riddle in which a fanciful question is answered by a pun.
2. n. A paradoxical, insoluble, or difficult problem; a dilemma: "the conundrum, thus far unanswered, of achieving full employment without inflation” ( Arthur M. Schlesinger, Jr.)

Century Dictionary and Cyclopedia

1. n. A conceit; a device; a hoax.
2. n. A riddle in which some odd resemblance is proposed for discovery between things quite unlike, or some odd difference between similar things, the answer often involving a pun.

Wiktionary

1. n. A difficult question or riddle, especially one using a play on words in the answer.
2. n. A difficult choice or decision that must be made.

GNU Webster's 1913

1. n. A kind of riddle based upon some fanciful or fantastic resemblance between things quite unlike; a puzzling question, of which the answer is or involves a pun.
2. n. A question to which only a conjectural answer can be made.

WordNet 3.0

1. n. a difficult problem