We introduce a new toy model for the study of glasses: the hard-matrix model. This may be viewed as a single particle moving on SO(N), where there is a potential proportional to the one-norm of the matrix. The ground states of the model are 'crystals' where all matrix elements have the same magnitude. These are the Hadamard matrices when N is divisible by four. Just as finding the latter has challenged mathematicians, our model fails to find them upon cooling and instead shows all the behaviors that characterize physical glasses. With simulations we have located the first-order crystallization temperature, the Kauzmann temperature where the glass would have the same entropy as the crystal, as well as the standard, measurement-time dependent glass transition temperature. Our model also brings to light a new kind of elementary excitation special to the glass phase: the rubicon. In our model these are associated with the finite density of matrix elements near zero, the maximum in their contribution to the energy. Rubicons enable the system to cross between basins without thermal activation, a possibility not much discussed in the standard landscape picture. We use these modes to explain the slow dynamics in our model and speculate about their role in its quantum extension in the context of many-body localization. © 2021 IOP Publishing Ltd and SISSA Medialab srl.