Mathematicians discover the worst way to hang a painting

In 1997, mathematician A. Spivak proposed a riddle: Is there a method to hang a painting using two nails and a string, such that removing either nail results in the painting falling? This question sparked a series of picture-hanging problems, which mathematicians have since explored. Retired computer scientist Tom Verhoeff first approached these issues at a grade school math camp, where campers experimented with real strings and carabiners.

In 2012, mathematicians published a preprint demonstrating that solutions exist for any "k-out-of-n" picture-hanging problem, where n represents the number of nails and the painting will fall if any k of the nails are removed. However, solutions to these problems often involve intricate string arrangements. At a workshop, Verhoeff and his team tackled the 2-out-of-4 problem, reducing the length of the shortest known solution from 80 to 58 wraps around the nails.

Later, with the help of Ph.D. student Jens Heuseveldt and a computer program, they further minimized the solution to 16 wraps. This exploration of complex picture-hanging scenarios connects to various mathematical fields, including group theory, knot theory, and graph theory. The seemingly trivial nature of the problem belies its connections to important mathematical concepts, and mathematicians find it intriguing.

Although the problem's framing might be amusing, the underlying mechanics have significant implications in other areas of mathematics, such as cryptography and voting theory. Verhoeff emphasizes that while the question of practical utility is unimportant, the value of play in learning cannot be understated. He argues that science journalism requires human expertise, time, effort, and creativity, and that funding this work is essential for fostering accurate and compelling scientific stories.