A bstract We find n ( n − 3) / 2-dimensional regions of the space of kinematic invariants, where all the solutions to the scattering equations (the core of the CHY formulation of amplitudes) for n massless particles are real. On these regions, the scattering equations are equivalent to the problem of finding stationary points of n − 3 mutually repelling particles on a finite real interval with appropriate boundary conditions. This identification directly implies that for each of the ( n − 3)! possible orderings of the n − 3 particles on the interval, there exists one stable stationary point. Furthermore, restricting to four dimensions, we find that the separation of the solutions into k ∈ {2 , 3 , . . . , n − 2} sectors naturally matches that of permutations of n − 3 labels into those with k − 2 descents. This leads to a physical realization of the combinatorial meaning of the Eulerian numbers.