Faulty picture-hanging improved

TL;DR

This paper revisits the classic picture-hanging puzzle, providing simplified proofs for its solvability and polynomial bounds on solutions, thereby enhancing understanding of the problem's theoretical foundations.

Contribution

It offers simplified proofs of the solvability and polynomial bounds for the k-out-of-n picture-hanging problem, improving upon previous complex arguments.

Findings

All reasonable picture-hanging puzzles are solvable.

Solution size can be bounded polynomially in the number of nails.

Simplified proofs lead to better bounds on solution complexity.

Abstract

A picture-hanging puzzle is the task of hanging a framed picture with a wire around a set of nails in such a way that it can remain hanging on certain specified sets of nails, but will fall if any more are removed. The classical brain teaser asks us to hang a picture on two nails in such a way that it falls when any one is detached. Demaine et al (2012) proved that all reasonable puzzles of this kind are solvable, and that for the $k$-out-of-$n$ problem, the size of a solution can be bounded by a polynomial in $n$. We give simplified proofs of these facts, for the latter leading to a reasonable exponent in the polynomial bound.