Mathematicians Report New Solutions to Classic Picture-Hanging Puzzle

Mathematicians have found new solutions to a long-studied puzzle about how to hang a picture on a wall using a loop of rope and multiple nails so that the picture falls if any one nail is removed, according to a report by Scientific American.

The puzzle, a recreational problem with roots in topology and group theory, asks how a string can be wound around several pegs in such a way that the arrangement remains stable only as long as every peg stays in place. Removing a single peg is meant to cause the entire structure to collapse, regardless of which peg is chosen.

According to Scientific American, researchers have now identified new configurations that satisfy this condition, including what the publication describes as particularly inefficient or convoluted arrangements—effectively identifying some of the most impractical ways to hang a painting while still meeting the puzzle's mathematical requirements.

The problem is connected to concepts in abstract algebra, including the use of commutators—mathematical expressions that capture how the order of operations affects an outcome—to encode the winding pattern of the string around the nails. Such techniques have long been used by mathematicians to study braids, knots and other structures where sequence and interdependence matter.

Scientific American reported that the new findings expand the known set of solutions to the picture-hanging problem, offering fresh examples for a puzzle that has interested both professional mathematicians and enthusiasts of recreational math.

The publication did not specify the individual researchers or institutions behind the newly reported solutions in the excerpt reviewed for this article.