= Solution
In the <binary branching model of an electromagnetic shower>, after $n$ ideal equal-sharing generations, the model has $N_n=2^n$ particles of <energy> $E_n=E_0/2^n$, at depth $nX_0$. Stop the branching when $E_n\le E_c$. Thus
$$
\boxed{n_{\mathrm{stop}}=\left\lceil\log_2\frac{E_0}{E_c}\right\rceil
=\lceil\log_2(10^4)\rceil=14.}
$$
The continuous estimate is $13.29X_0$. At 13 generations the particles still have approximately $12.2\,\mathrm{MeV}$, whereas at 14 they have approximately $6.10\,\mathrm{MeV}$. The ideal branching model therefore suggests roughly $13$–$14X_0$ before multiplication stops.
This is not an accurate containment criterion. Real conversion and radiation distances fluctuate, <bremsstrahlung> <energy> sharing is broad rather than always one half, some energetic branches penetrate beyond the average, and particles below <Critical energy of an electromagnetic shower>[critical energy] still have stopping and absorption ranges. The simplified stopping depth is closer to its own model's shower-maximum scale than a depth containing all deposited <energy>. A <Particle calorimeter>[calorimeter] needs additional longitudinal depth to control leakage and event-to-event tails. These effects explain why approximately $25X_0$ can be required even though the equal-sharing estimate is smaller. The toy model does not predict the realistic <Electromagnetic shower>[shower] maximum or leakage fraction quantitatively.
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