= Solution
Assume the quoted <intensities> represent relative emitted <photon numbers>, with efficiency corrections implicit, as required for population-flow bookkeeping. The prominent branch obeys $I_1=I_2+I_4$, and $I_2=I_3$ identifies a serial two-photon branch. The energies obey $E_2+E_3\simeq E_4$ and $E_1+E_4\simeq E_5$ within their uncertainties. The weak fifth line directly spans D to A. Reversing which end contains the line-1 step and interchanging the serial line-2/line-3 order produces the four possibilities.
Take A as zero. A <gamma-ray energy-network fit> to the five <energy> differences gives the following approximate <excitation energies> in keV:
$$
\boxed{\begin{array}{c|rrr}
\text{arrangement}&\Delta E_{AB}&\Delta E_{AC}&\Delta E_{AD}\\\hline
\mathrm I&133.02&269.27&615.10\\
\mathrm{II}&133.02&478.85&615.10\\
\mathrm{III}&136.25&482.08&615.10\\
\mathrm{IV}&345.83&482.08&615.10
\end{array}}
$$
The line connections, in the order $1,2,3,4,5$, are respectively: I: $(B\to A,C\to B,D\to C,D\to B,D\to A)$; II: $(B\to A,D\to C,C\to B,D\to B,D\to A)$; III: $(D\to C,B\to A,C\to B,C\to A,D\to A)$; IV: $(D\to C,C\to B,B\to A,C\to A,D\to A)$.
For I and II, the B population entering through lines 2 or 3 and line 4 exits through line 1. For III and IV, line 1 feeds C and its <intensity> splits into the direct line 4 and serial lines 2/3. The small direct D-to-A branch bypasses these intermediate balances. The central measured energies do not obey every sum exactly; all four fitted arrangements have $\chi^2\simeq0.81$ for two <energy> constraints. Assigning the weakest line's central <energy> as an exact <excitation energy> would discard the more precise cascade measurements.
\Image[/past-exam-of-the-particle-and-nuclear-physics-course-of-the-university-of-cambridge/2014/paper-1-four-level-arrangements.png]
{title=Original reconstructed solutions I–IV, with all five observed line numbers attached to calculated transitions; excitation energies are fitted, not a copy of the source diagram}
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