Problem, n.
Etymology: [F. problème, L. problema, fr. Gr. Parable. ]
1. A question proposed for solution; a matter stated for examination or proof; hence, a matter difficult of solution or settlement; a doubtful case; a question involving doubt. Bacon.
2. (Math.)
Anything which is required to be done; as, in geometry, to bisect a line, to draw a perpendicular; or, in algebra, to find an unknown quantity.
Note: Problem differs from theorem in this, that a problem is something to be done, as to bisect a triangle, to describe a circle, etc.; a theorem is something to be proved, as that all the angles of a triangle are equal to two right angles. Plane problem (Geom.), a problem that can be solved by the use of the rule and compass. -- Solid problem (Geom.), a problem requiring in its geometric solution the use of a conic section or higher curve.
Shades of meaning
Fernald has no entry of its own for problem; he lists it under riddle.
Where he writes about it:
Conundrum, a word of unknown origin, signifies some question or statement in which some hidden and fanciful resemblance is involved, the answer often depending upon a pun; an enigma is a dark saying; a paradox is a true statement that at first appears absurd or contradictory; a problem is something thrown out for solution; puzzle (from oppose) referred originally to the intricate arguments by which disputants opposed each other in the old philosophic schools. — from riddle
The riddle is an ambiguous or paradoxical statement with a hidden meaning to be guessed by the mental acuteness of the one to whom it is proposed; the riddle is not so petty as the conundrum, and may require much acuteness for its answer; a problem may require simply study and scholarship, as a problem in mathematics; a puzzle may be in something other than verbal statement, as a dissected map or any perplexing mechanical contrivance. — from riddle