A Note on a Picture-Hanging Puzzle

Radoslav Fulek; Sergey Avvakumov

A Note on a Picture-Hanging Puzzle

Combinatorics 2018-12-20 v2 Discrete Mathematics

Authors: Radoslav Fulek , Sergey Avvakumov

Abstract

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 $n2^{\sqrt{\log_2 n}}$ . 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.

Cite

@article{arxiv.1812.06335,
  title  = {A Note on a Picture-Hanging Puzzle},
  author = {Radoslav Fulek and Sergey Avvakumov},
  journal= {arXiv preprint arXiv:1812.06335},
  year   = {2018}
}

Comments

We have learned that a stronger lower bound, that is also tight, was published before in: P. Gartside and S. Greenwood, Brunnian links, Fundamenta Mathematicae 193, (2007), url: https://eudml.org/doc/282667

A Note on a Picture-Hanging Puzzle

Abstract

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