In the quantum three-body problem, There are three possible models: 1. Each particle interacts independently with the two others, forming three interacting pairs. Pairwise interactions refer to this model. 2. Each particle interacts with the centre of mass of the two others, called the pure three-body interaction. 3. Each particle interacts simultaneously with the two others and with the centre of mass of the two others. This latest is called the full-three-body interaction, which combines pairwise and pure three-body interactions. Therefore, knowing which of these models prevails in a given physical context is essential. In this paper, we choose the integrable Calogero Sutherland Hamiltonian, and we explicitly write the wave functions of the ground states in each of these modellings for three fermions and three bosons on a ring. These wave functions reveal regions of the null or quasi-null probability of the presence of particles. This theoretical distribution of probability is to be compared with observations to discover which of these models prevails in the observed physical context. As a physical context, we suggest cold atoms’ ring-shaped optical lattices. In addition, we clarify several points about the relation between trigonometric and inverse square interactions on a line and a ring.