How was the critical mass calculated for Little Boy? Little Boy was detonated in 1945, 6 years after nuclear chain reactions began to be experimentally proven and Leo Szilard and Albert Einstein warned Roosevelt of the dangers of Germany developing nuclear weapons. How was the critical mass calculated for Little Boy? To a lay man, it seems like trial and error was problematic, and that it had to rely on theoretical calculation.
 A: Any estimate of a critical mass is going to require a mixture of theory and experimental data. Nuclear physics was in a very primitive stage in 1945, e.g., the nuclear shell model by Maria Goeppert Mayer et al. was not really developed in any mathematical detail until about 4 years later. The WP article "Critical mass" actually has quite a nice treatment of this topic. In their notation, there are several parameters that are needed:
s = the number of scatterings per fission
$\sigma$ = total scattering cross-section
f = a factor involving geometry and modeling of things like diffusion
The factors $s$ and $\sigma$ had to be determined experimentally. The factor $f$ can be roughly estimated fairly easily, but to get an accurate estimate required much more difficult calculations. Because there were many unknowns involved and many surprises discovered in experiments, it was necessary to go through a cycle of theory and experiment, which started with Chicago Pile-1 in 1942 and continued beyond 1945.
The ability to do lab experiments was a huge aid in the Manhattan Project. The inability to do them was a big difficulty in the later development of the hydrogen bomb.
A: The critical mass can be estimated as follows:
You take a subcritical mass of given shape and strike it with a brief pulse of fast neutrons. This induces a burst of fission activity inside the subcritical mass, the size of which is measured. You repeat the experiment many times with a slightly larger pulse of neutrons each time and record the responses, including the time required for the fission activity to die away after each input pulse. 
Charting out the data, you then extrapolate to the "untestable" case where the induced fission burst does not die away i.e., the chain reaction becomes self-sustaining, and from this you can estimate what the critical mass should be without having to actually assemble a critical mass in the lab. 
