Picture-Hanging Puzzles

TL;DR

This paper introduces a mathematical construction for picture-hanging puzzles that ensures the picture falls if any k out of n nails are removed, linking the problem to monotone Boolean functions and discussing complexity considerations.

Contribution

It characterizes all possible monotone Boolean functions for picture-hanging puzzles and analyzes the complexity of constructions.

Findings

Construction ensures the picture falls when any k nails are removed.

Characterization of all monotone Boolean functions for the puzzle.

Exponential complexity is necessary for general functions.

Abstract

We show how to hang a picture by wrapping rope around n nails, making a polynomial number of twists, such that the picture falls whenever any k out of the n nails get removed, and the picture remains hanging when fewer than k nails get removed. This construction makes for some fun mathematical magic performances. More generally, we characterize the possible Boolean functions characterizing when the picture falls in terms of which nails get removed as all monotone Boolean functions. This construction requires an exponential number of twists in the worst case, but exponential complexity is almost always necessary for general functions.