More practically (and specifically ruled out in the question): you can see when your friend "starts" the exam (picks up his pencil) and when he ends (gets up fr

More practically (and specifically ruled out in the question): you can see when your friend "starts" the exam (picks up his pencil) and when he ends (gets up from his seat). Also it probably takes him at least 10 minutes to read the whole test and figure out the answers. This makes it easy mode: friend takes 10 minutes to read test and get answers in his head, start the timer (0 seconds) at the 10 minute mark. Friend "starts" sometime in the next 34 minutes. 34 minutes is 2048 seconds or 11 bits of info. Then "finishes" between 44 minutes and 78 minutes: another 34 minute window and another 11 bits of info. 22 of 23 questions answered. Friend coughs on way out if last answer is true. Boom, all questions answered. Yes, OP said we can't do this, but it is more practical than memorizing 5400 bit vectors, doesn't assume the friend can instantly finish the test, gives the friend some lead time to actually do some binary conversions, and ensure the correct times down to the second.