We know the meaning of 振り in 手振り身振り is "imitation," combined with words 手 and 身 to form the meaning of "use of gestures and body languages to mimic."
But in the word 無茶振り, 振り does not seem to mean "imitation". How should I interpret this 振り?
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振る has many meanings including:
10. to bring up a topic; to lead to a topic
12. to set up a joke for somebody else
These were derived from the more basic meaning of 振る, "to assign/allocate (a task)". 話を振る/話題を振る is a common idiom.
The noun 振り also has many meanings, one of which is "lead-in (to a joke, topic, question, etc.)". For example, in a variety show, if someone says the first part of a running joke and looks at someone else, that's a 振り. The word is typically used by comedians and TV hosts. 無茶振り is a word to refer to the 無茶 ("extremely difficult; unreasonable") type of 振り, which may result in an awkward silence or "eh?"
The word 無茶振り became so popular that it gained broader meanings outside the field of owarai. Now it can also mean "to demand something seemingly impossible (and see what happens)", "to assign a very difficult task", etc.