  For integers $r,t\geq2$ and $n\geq1$ let $f_r(t,n)$ be the minimum, over all
factorizations of the complete $r$-uniform hypergraph of order $n$ into $t$
factors $H_1,\dots,H_t$, of $\sum_{i=1}^tc(H_i)$ where $c(H_i)$ is the number
of maximal cliques in $H_i$. It is known that $f_2(2,n)=n+1$; in fact, if $G$
is a graph of order $n$, then $c(G)+c(\overline G)\geq n+1$ with equality iff
$\omega(G)+\alpha(G)=n+1$ where $\omega$ is the clique number and $\alpha$ the
independence number. In this paper we investigate $f_r(t,n)$ when $r>2$ or
$t>2$. We also characterize graphs $G$ of order $n$ with $c(G)+c(\overline
G)=n+2$.
