Let $X_1,\dots, X_n$ be independent integers distributed uniformly on $\{1,\dots, M\}$, $M=M(n)\to\infty$ however slow. A partition $S$ of $[n]$ into $ν$ non-empty subsets $S_1,\dots, S_ν$ is called perfect, if all $ν$ values $\sum_{j\in S_{\a}}X_j$ are equal. For a perfect partition to exist, $\sum_j X_j$ has to be divisible by $ν$. For $ν=2$, Borgs et al. proved, among other results, that, conditioned on $\sum_j X_j$ being even, with high probability a perfect partition exists if $κ:=\lim \tfrac{n}{\log M}>\tfrac{1}{\log 2}$, and that w.h.p. no perfect partition exists if $κ<\tfrac{1}{\log 2}$. We prove that w.h.p. no perfect partition exists if $ν\ge 3$ and $κ<\tfrac{2}{\log ν}$. We identify the range of $κ$ in which the expected number of perfect partitions is exponentially high. We show that for $κ> \tfrac{2(ν-1)}{\log[(1-2ν^{-2})^{-1}]}$ the total number of perfect partitions is exponentially high with probability $\gtrsim (1+ν^2)^{-1}$.