The classic picture-hanging puzzle asks for a way to wrap a wire around two nails such that the picture hangs as long as no nails are removed, but falls as soon as any one nail is removed. Mathematicians have generalized this puzzle in several steps:

• The picture hangs on N nails and falls when any one nail is removed (the 1-of-N puzzles).
• The picture hangs on N nails and still hangs when fewer than K nails are removed, but falls when any K nails are removed (the K-of-N puzzles).
• Given is a monotonic boolean function B^N -> { Hang, Fall }, not constant Hang, with Fall < Hang. Wrap the wire such that the picture hangs/falls according to f(b_1, b_2, …, b_N) where b_i = “nail i is removed”.

In 2014, Erik Demaine et al. wrote a paper that solves all these problems and they give some bounds on the complexity of solutions: https://arxiv.org/abs/1203.3602.
Earlier this year, while using this problem in two outreach activities, I improved some of those bounds. In particular, I also constructed an optimal solution for the 2-of-4 problem. (Demaine’s paper gives a solution with 80 wrappings.)

In this colloquium, I will explain the problem, let you play with a physical setup, explain the algebraic theory, and summarize my results.
Earlier this month, an article about my results appeared in Scientific American: https://www.scientificamerican.com/article/mathematicians-discover-the-worst-way-to-hang-a-painting/.
My paper can be found here: https://arxiv.org/abs/2605.27617.