A few years ago, I took part in an interview for a library job where one of the questions was, "Tell us about a situation in which there was no possible solution-- what you did in response?" I thought of the most impossible situation I ever had, where a patron who spoke English as a second language wanted a book about the Headless Horseman and followed the plot-line of The Legend of Sleepy Hollow, and yet was not Washington Irving and not in the children's section because her 13-year-old son was far too advanced for children's books. After doing an extensive search, another librarian and I figured out that the woman may have read the book in translation in her home country and thought the translator was the author.
There were copies of The Legend of Sleepy Hollow in both the children's and adult sections of the library, but when it was apparent that none of the librarians on staff could produce the book she described that was not written by Washington Irving, the woman stormed off with her embarrassed 13-year-old son in tow. It felt like an impossible situation.
Later on, it occurred to me that I hadn't answered the interview question correctly. What the interviewers may have wanted to hear was a chirpy, "There are no impossible situations! Here is an example of a situation that might have seemed impossible to someone else, but I was able to solve it using the know-how of my library science degree plus years of experience."
One of the major challenges a library staff person can do is to figure out what a patron really wants when it's different from what a patron says s/he wants. In that same job interview, I talked about another patron who wanted information on an ancient form of Judaism, and I figured out that what she actually wanted was Janism, an offshoot of Hinduism.
Have you ever dealt with the impossible questions, either in your work environment or in an interview? What did you say? What do you wish you had said?