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Filling a room


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#1 BMAD

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Posted 10 October 2013 - 05:43 PM

You have an empty room, and a group of people waiting outside the room. At each step, you may either get one person into the room, or get one out. Can you make subsequent steps, so that every possible combination of people is achieved exactly once?


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#2 austinm

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Posted 11 October 2013 - 04:45 AM

Spoiler for Recursively:

Edited by austinm, 11 October 2013 - 04:50 AM.

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#3 austinm

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Posted 11 October 2013 - 01:30 PM


 

Proof that the method works is inductive on n. (Left as exercise to reader.)

 

By the way, I wasn't just being coy here. The proof is a bit interesting--contains one or two knots that have to be thought-through carefully--and is worth taking five minutes to lay out.


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