= Solution
Measure ring centers and <radii> against the printed scale, rather than comparing raw squared <radii>. A representative reading of the four rings is $R\simeq6.0,7.0,10.6,13.2\,\mathrm{mm}$, with broad photographic lines. For a flat plate at distance $L$,
$$
2\theta=\arctan(R/L),\qquad
\sin^2\theta=\frac12\left(1-\frac{L}{\sqrt{L^2+R^2}}\right).
$$
With $L=16\,\mathrm{mm}$ the $\sin^2\theta$ ratios are approximately $1:1.33:2.67:3.67$, giving
$$
\boxed{111,\quad200,\quad220,\quad311;\qquad\text{face-centered cubic lattice}.}
$$
For identical atoms at the four conventional FCC positions,
$$
F_{hkl}=f\{1+(-1)^{h+k}+(-1)^{h+l}+(-1)^{k+l}\}.
$$
The <structure factor> is $4f$ for indices all even or all odd, and zero for mixed parity. Thus, for example, 100, 110 and 210 are <systematic absences>. The allowed squared-index sums begin $3,4,8,11,12,16,\ldots$. Only the first four rings fit within the displayed field; 222 and later allowed reflections are not forbidden merely because they are outside the image. The central shadow and its holder are not extra <diffraction> peaks.
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