Let $X_1,\ldots, X_n$ be independent integers distributed uniformly on [M], $M\ge 2$. A partition S of [n] into $\nu$ non-empty subsets $S_1,\ldots, S_{\nu}$ is called perfect if all $\nu$ values $\sum_{j\in S_{\alpha}}X_j$ are equal. For a perfect partition to exist, $\sum_j X_j$ has to be divisible by $\nu$. In 2001, for $\nu=2$, Christian Borgs, Jennifer Chayes, and the author proved that, conditioned on $\sum_j X_j$ being even, with high probability a perfect partition exists if $\kappa\;:\!=\; \lim {{n}/{\log M}}>{{1}/{\log 2}}$, and that with high probability no perfect partition exists if $\kappa<{{1}/{\log 2}}$. Responding to a question by George Varghese, we prove that for $\nu\ge 3$ with high probability no perfect partition exists if $\kappa<{{2}/{\log \nu}}$, which is twice as large as the naive threshold $1/\log 3$ for $\nu=3$. We identify the range of $\kappa$ where the expected number of perfect partitions is exponentially high. We show that for $\kappa> {{2(\nu-1)}/{\log[(1-2\nu^{-2})^{-1}]}}$ the total number of perfect partitions is exponentially high with probability $\gtrsim (1+\nu^2)^{-1}$, i.e. below $1/\nu$, the limiting probability that $\sum_j X_j$ is divisible by $\nu$.