A simple quasi-lattice model, in which no pair of neighbouring sites may be occupied, can be solved exactly. The `lattice' consists of triangles of sites connected at corners, and can be split into three equal sublattices, one of which is preferentially occupied when the overall density exceeds a certain value. This model reproduces the behaviour of the continuum rigid-sphere model much more closely than do lattice models possessing only two sublattices. There is more than one configuration that makes the free energy a local minimum, and we have to select the most stable one. When this is done, the transition is first order, and the isotherm consists of two distinct portions, closely resembling what is obtained for the rigid-sphere model.