We study the asymptotics of a Markovian system of N ≥ 3 particles in [0, 1]
in which, at each step in discrete time, the particle farthest from the current centre of mass is removed and replaced by an independent U[0, 1]
random particle. We show that the limiting configuration contains N − 1 coincident particles at a random location ξ
∈ [0, 1]
. A key tool in the analysis is a Lyapunov function based on the squared radius of gyration (sum of squared distances) of the points. For d = 1, we give additional results on the distribution of the limit ξ
, showing, among other things, that it gives positive probability to any nonempty interval subset of [0, 1], and giving a reasonably explicit description in the smallest nontrivial case, N = 3.