We have determined the arrangement of magnetic moments (spins) in manganese oxide (MnO) crystals by observing the magnetic interactions between individual pairs of atoms. Spins in solids underpin a wide variety of physical phenomena of important technological application, including (for example) magnetic recording media. It was known in the 1940s that magnetic structure could cause a beam of neutrons to scatter into a diffraction pattern of intense isolated spots. By analysing the position and brightness of each spot, one can deduce the average arrangement of spins within the sample. The first successful experiment – the discovery in 1949 of the basic magnetic structure of MnO – led in part to the 1994 Nobel prize in Physics. Despite the considerable development of the technique since, the full magnetic structure of MnO has remained elusive. The problem has been that there are a number of different possible structures that would give rise to precisely the same diffraction pattern. In our work, we showed how the very weak but persistent scattering between the spots holds vital information about the interactions between individual pairs of atoms. This in turn describes the actual arrangement of spins in the material, which we found has some unexpected complexities.