In ADM numerical relativity, I take it that initial data is given as a 3-dimensional $g_{ij}$ configuration slice frozen in time (eg. BL, Misner solutions). The time evolution is to be calculated step by step via
$$ \frac{d}{dt}g_{ij} = \frac{2N}{\sqrt g}(\pi_{ij} - \frac{1}{2}\pi g_{ij}) + N_{i;j} + N_{j;i}$$
However, in the exposition provided here, all the quantities used for calculating $\frac{d}{dt}g_{ij}$ has to be pre-calculated using the 4-dimensional metric $^{(4)}g_{\mu\nu}$ and Christoffel connections $^{(4)}\Gamma^i_{jk}$.
This seems like a catch-22.
Since the initial data is presumably only for a 3-dimensional space frozen in time, how would we get the rest of the data to pre-construct the the full 4-dimensional $g_{\mu\nu}$ over time, in order to calculate $^{(4)}g_{\mu\nu}$, $^{(4)}\Gamma^i_{jk}$, and finally back to calculating $\frac{d}{dt}g_{ij}$?
As you can see in the time evolution, you need the conjugate momenta $\pi^{i,j}$. Purely in terms of free parameters, $g_{\mu,\nu}$ has 10, while $g_{i,j}$ and $\pi^{i,j}$ have 6 each, giving 12. The two extra parameters are "eaten" by your lapse function and shift vector if I recall correctly (although on that part I am more fuzzy). So as usual with initial value problems, you also need the first derivative at the start time in order to get anything sensible.