At some point during your childhood, you probably played with a popular sliding puzzle game that originated in the 1880s consisting of tiles in a grid—most likely a 3x3 grid (an 8-puzzle) or 4x4 grid (a 15-puzzle). The concept was simple: get the tiles in numerical order by sliding the tiles, utilizing the empty space. It was frustrating. It was addicting.  It was challenging. Now, physicists are using it as a way to try and understand how magnets work, which is one of the most complex puzzles. The Not So Simple Science Behind Permanent Magnets Ferromagnetism is the science behind how permanent magnets—such as the ones that stick to your refrigerator—work. With this mechanism, the spins of electrons align and generate a magnetic field. Certain metals, including cobalt, iron, and nickel, have electrons that move around freely. The electrons themselves also have magnetic moments, but understanding how these moments align has to be determined by calculating quantum interactions between them. Yi Li, a physicist at Johns Hopkins University, and graduate students Eric Bobrow and Keaton Stubis used the mathematics behind the 15-puzzle to expand the theorem that describes itinerant ferromagnetism. Their findings, published in Physical Review B, explain a more realistic system for how magnets work. How to Explain Ferromagnetism There are two constraints that electrons in metal have to abide by: first, the electrons must repel one another because they are negatively charged, and second, they have to obey the Pauli exclusion principle. This principle states that no two particles can occupy the same quantum state. While electrons with the same spin already can’t do this, two electrons with opposite spins could occupy the same quantum state. Ferromagnetism is the easiest solution to these constraints. Here, the electrons stay separated and align their spins. Where physicists are stuck? Finding a detailed model of how the pattern of aligned spins (ferromagnetism) results from the quantum interactions between electrons. The Nagaoka-Thouless Theorem In the 1960s, two physicists—Yosuke Nagaoka and David Thouless—independently derived and published proofs that explained why electrons align and generate a ferromagnetic state. This was the first explanation for why electron spins should align. Their theorem is based on an idealized system of electrons in a lattice. Extending the Nagaoka’s Theorem Hal Tasaki, a physicist at Gakushuin University in Japan, extended the theorem in 1989 by discovering that it would apply as long as a lattice has connectivity. The connectivity condition of a lattice is satisfied if you are able to create all configurations of a spin while maintaining the number of spin-up and spin-down electrons in a lattice with one moving hole. Tasaki wasn’t able to determine if the connectivity condition would be satisfied in lattices other than square, triangle, and three-dimensional cubic. The Case for the 15-Puzzle Bobrow and Stubis realized the honeycomb lattice Li was using in an attempt to answer Tasaki’s question closely resembled the 15-puzzle if the numbers on the tiles were instead up or down spins. The puzzle then represents a Nagaoka ferromagnet—the hole moves through the tiles, or in this instance, a lattice of electrons. Using solutions for all lattices of the 15-puzzle from mathematician Richard Wilson, the researchers determined that the connectivity condition is satisfied for most of the lattices. This discovery gets physicists one step closer to a more complete model of itinerant ferromagnetism. But magnetism doesn’t stop there: there are several different ways to classify magnetism. If you’re interested in learning more about the physics behind magnets, or about magnetism in general, follow our magnets in the news blog or subscribe to our newsletter.