  Let $\mathbb{T}^d_N$, $d\ge 2$, be the discrete $d$-dimensional torus with
$N^d$ points. Place a particle at each site of $\mathbb{T}^d_N$ and let them
evolve as independent, nearest-neighbor, symmetric, continuous-time random
walks. Each time two particles meet, they coalesce into one. Denote by $C_N$
the first time the set of particles is reduced to a singleton. Cox [6] proved
the existence of a time-scale $\theta_N$ for which $C_N/\theta_N$ converges to
the sum of independent exponential random variables. Denote by $Z^N_t$ the
total number of particles at time $t$. We prove that the sequence of Markov
chains $(Z^N_{t\theta_N})_{t\ge 0}$ converges to the total number of partitions
in Kingman's coalescent.
