= Solution
The supplied average retained-energy fraction in one <elastic collision> is
$$
f=\frac{11.18^2+0.94^2}{(11.18+0.94)^2}=0.8569149.
$$
The <mean-energy neutron moderation count> follows by iterating the stated mean-energy model for <neutron moderation>, giving $E_n=f^nE_0$. Equivalently, take logarithms to solve the geometric decrease:
$$
n=\frac{\ln(E_n/E_0)}{\ln f}
=\frac{\ln[0.025/(2\times10^6)]}{\ln(0.8569149)}
=117.847.
$$
Hence the estimated integer count is
$$
\boxed{n=118\ \text{collisions}.}
$$
After $117$ collisions the predicted mean energy is about $0.0285$ eV; after $118$ it is about $0.0244$ eV, bracketing the target. This is a mean-energy estimate using the given average retention factor. Individual scattering angles and energy losses fluctuate, so a specified single <neutron> is not guaranteed to reach the target on exactly its $118$th collision. An estimate based on the average logarithmic energy decrement is a different statistical convention.
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