Send a secret hour using four books on a shelf

Question

Two teammates plan ahead. One of them learns the meeting time, some hour of the day, 1 through 24, and must pass it silently to the other. The only channel: a shelf holding four distinct books, which the first teammate may arrange in any left-to-right order. No notes, no bent pages, no marks, just the order of the four books.

Can they always communicate the exact hour? Show how.

Accepted Answer

Yes, and there is exactly enough room to do it, no more, no less. Count the arrangements. Four distinct books can be lined up in 4 	imes 3 	imes 2 	imes 1 = 24 different orders. There are exactly 24 hours. So there is a perfect one-to-one match between orderings and hours: assign each of the 24 possible orders its own hour, agree on that master list in advance, and you are done. The sender arranges the books in the order that stands for the target hour; the receiver reads the order and looks it up. A clean way to build the list (so you don't memorize 24 rows). Rank the four books by an agr…

Send a secret hour using four books on a shelf

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