When I was taking an advanced fluid mechanics course for my undergrad degree, most of the class pestered the Prof endlessly for a take home midterm exam. He did not like take home tests, then begrudgingly agreed with one caveat - no one could spend more than 6 hours working on the exam. I got to the last exam question, ran calculations until I was at the 6 hour limit and realized that the problem was unsolvable, which was my answer.
When the Prof discussed the exam results, he revealed that he had indeed put an unvsolvable problem on the test, only two of us got the question correct and everyone else spent a ridiculous amount of hours trying to figure out a solution. No one asked or a take home final.