  We discuss the phase diagram of the balls in boxes model, with a varying
number of boxes. The model can be regarded as a mean-field model of simplicial
gravity. We analyse in detail the case of weights of the form $p(q) =
q^{-\beta}$, which correspond to the measure term introduced in the simplicial
quantum gravity simulations. The system has two phases~: {\em elongated} ({\em
fluid}) and {\em crumpled}. For $\beta\in (2,\infty)$ the transition between
these two phases is first order, while for $\beta \in (1,2]$ it is continuous.
The transition becomes softer when $\beta$ approaches unity and eventually
disappears at $\beta=1$. We then generalise the discussion to an arbitrary set
of weights. Finally, we show that if one introduces an additional kinematic
bound on the average density of balls per box then a new {\em condensed} phase
appears in the phase diagram. It bears some similarity to the {\em crinkled}
phase of simplicial gravity discussed recently in models of gravity interacting
with matter fields.
