  We study $\mathcal{N}=4$ SYM at non-integer number of colours. By varying $N$
we can continuously follow states all the way from $N=\infty$ where
integrability reigns to finite $N$ where quantum gravity effects dominate. As
an application we consider classically $1/16$ BPS states. Quantum mechanically,
these states are generically non-supersymmetric but some special states - at
special values of $N$ - become super-symmetric at the quantum level as well.
They are the so-called quantum black hole states studied recently using
cohomology. We write down the form of the lightest BH state at $N=2$ - and
follow it in $N$, both at weak coupling and - more speculatively - at strong
coupling as well. At weak coupling this state has protected dimension
$\Delta=19/2$ at $N=2$ and becomes a triple trace made out of Konishi and two
light BPS operators at infinite $N$ with $\Delta=19/2+12\lambda+\dots$. At
strong coupling we suspect it becomes a quadruple trace with dimension $\Delta
\simeq 19/2+\text{integer}$.
