puzzle over vs cogitate

puzzle over and cogitate both are verbs.

Word	Noun	Adjective	Verb	Adverb
puzzle over	No	No	Yes	No
cogitate	No	No	Yes	No

As verbs, cogitate is a hypernym of puzzle over; that is, cogitate is a word with a broader meaning than puzzle over:

puzzle over: try to solve
cogitate: use or exercise the mind or one's power of reason in order to make inferences, decisions, or arrive at a solution or judgments

Other hypernyms of puzzle over include cerebrate, think.

puzzle over (verb)	cogitate (verb)
try to solve	use or exercise the mind or one's power of reason in order to make inferences, decisions, or arrive at a solution or judgments
	consider carefully and deeply; reflect upon; turn over in one's mind