You have 13 reels of magnetic tape, one empty reel, and a machine that will wind tape from a full reel to an empty one, reversing its direction. You need all 13 tapes reversed on their original reels. Show how this can be done, or prove that it’s impossible.
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It’s impossible. Each of the 13 reels must go through an even number of transfers in order to finish with its original tape. (Likewise, the empty reel must go through an even number of transfers to remain empty.) But each of the 13 tapes must be transferred an odd number of times to end up reversed. So the total number of transfers must be both even and odd, an impossibility.
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