  The discovery of magnetic bistability in Mn$_{12}$ more than 20 years ago
marked the birth of molecular magnetism, an extremely fertile interdisciplinary
field and a powerful route to create tailored magnetic nanostructures. However,
the difficulty to determine interactions in complex polycentric molecules often
prevents their understanding. Mn$_{12}$ is an outstanding example of this
difficulty: although it is the forefather and most studied of all molecular
nanomagnets, an unambiguous determination of even the leading magnetic exchange
interactions is still lacking. Here we exploit four-dimensional inelastic
neutron scattering to portray how individual spins fluctuate around the
magnetic ground state, thus fixing the exchange couplings of Mn$_{12}$ for the
first time. Our results demonstrate the power of four-dimensional inelastic
neutron scattering as an unrivaled tool to characterize magnetic clusters.
