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I apologize if the question is too vague. I thought I could provide some background to clarify what I mean by separate and arrange.

I was chatting with a friend one day and he had 4 keys and could pick out two of them by flipping the first one on the ring so that he could feel the teeth of that key and the second key would be in the middle of the three keys remaining.

He recently got another key and now has 5. The same technique doesn’t work because there are 4 keys remaining after feeling the teeth of the first.

We were trying to think of a process that would allow him to pick out the same two keys in the right order.

Also, I was wondering if there was a way to frame this problem with ‘n’ total keys and having to pick out ‘m’ of them.

Alexander Julian Ty
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  • I don't quite understand what you mean by "pick out." With 4 keys, say they are a,b,c, and d in cyclic order. Then he can separate {a,c} for {b,d} with this technique, but he doesn't know which pair is which, does he? Also, I don't see how he can separate {a,b} for {c,d}. Can you clarify the question for me? – saulspatz Dec 23 '17 at 19:25

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If he flips the first and third keys, going one way around there is only one key between the flipped ones and the other way there are two. Define the first flipped key by having one normal before it and two after.

Ross Millikan
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