A Note on a Picture-Hanging Puzzle

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 $n{2}^{\sqrt{{\log }_{2}n}}$. 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.

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

Cited by in corpus (1)

• Faulty picture-hanging improved