Is there a word that is equivalent to the phrase 'intellectual challenge'?
There are many that go around this idea (paradox, theory, riddle, quandary, etc...)
However, for the aspect of "challenge," specifically, I would propose:
1a : an intricate and difficult problem
1b : a question or problem having only a conjectural answer
This especially hits on the "intellectual" nature, since conjectural means that whatever answer you come up with cannot be 'physically' proven or tested: you just have to be right.