  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.

Combinatorics;Discrete Mathematics