A Note on a Picture-Hanging Puzzle

12/15/2018

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by   Radoslav Fulek, et al.

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In the picture-hanging puzzle we are to hang a picture so that the string loops around n nails and the removal of any nail results in a fall of the picture. We show that the length of a sequence representing an element in the free group with n generators that corresponds to a solution of the picture-hanging puzzle must be at least n2^√(_2 n). In other words, this is a lower bound on the length of a sequence representing a non-trivial element in the free group with n generators such that if we replace any of the generators by the identity the sequence becomes trivial.