Looking for single word definition for a question or riddle that seemingly has no answer. Designed to confuse. Not paradox or conundrum. I.e "What is the sounds of on hand clapping?"
That would be a Zen (short) story ;-) . See:
a seemingly absurd or contradictory statement or proposition which when investigated may prove to be well founded or true: the uncertainty principle leads to all sorts of paradoxes, like the particles being in two places at once
a confusing and difficult problem or question: one of the most difficult conundrums for the experts
a person or thing that is mysterious or difficult to understand: Madeleine was still an enigma to him
you could call it a catch-22 defined in the dictionary as
- a frustrating situation in which one is trapped by contradictory regulations or conditions.
- any illogical or paradoxical problem or situation; dilemma.
- a condition, regulation, etc., preventing the resolution of a problem or situation; catch.
I recognize that this does not actually answer the question, but… I think there is a modern tendency to insist on having a word for something, at any cost. I suggest that you allow the possibility of having to use more than one word.
For the definitive feature of the expression, I would use “paradox” or “incoherent” (or possibly “internally inconsistent”)… or just “meaningless”.
I think here I would use “incoherent question” — given your desire for conciseness. With the luxury of using enough words: “a question built around an internally inconsistent concept”.
(The term “meaningless” includes things that are quite without meaning, which is not useful here.)