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gjm comments on Harry Potter and the Methods of Rationality discussion thread, part 20, chapter 90 - Less Wrong
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Made trickier by the fact that the information needs to be reliably checkable by Harry. And preferably not something he might already have known but mostly-forgotten.
He needs, in other words, a problem in class NP but not P. (Where P is "Potter" rather than "polynomial".)
"The location and description of a specific item in your muggle bedroom" or "Chapter, author, title and quotation from a book that you read but I did not" and "details on your family that you haven't yet told me about" seem like places to start. (The bedroom/family ones require he didn't get any such information over the Christmas break, but from the sounds of it he didn't.)