A Note on a Picture-Hanging Puzzle

TL;DR

This paper establishes a lower bound on the complexity of solutions to a classic picture-hanging puzzle, showing that the sequence length must grow significantly with the number of nails to ensure the puzzle's properties.

Contribution

It provides a mathematical lower bound on the sequence length needed for solutions in the picture-hanging puzzle, linking group theory to puzzle complexity.

Findings

Lower bound on sequence length is at least n*2^{√(log₂ n)}

Solutions require exponentially long sequences as number of nails increases

Mathematical connection between free groups and puzzle complexity

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.