Finding quantum numbers for particle in a 3D box

[A$APFowler]

I'm using the Modern Physics by Tipler (6th edition) book.
In sec 7.1 it talks about the first excited state being either E_(112 ) E_(121 ) E_(112).

My question is what is the process of finding the n_(1),n_(2),n_(3) quantum numbers ? How i understand you pick random values and from their find the order of energy levels. Can you give the process if would take of finding let's say the 4th excited state for both a cubic and non cubic box.

Thank you for your time.

PS: I apologize if my format is incorrect this is my fist post. The "_" represent subscript.

 

[clamtrox]

Well, if the energy dependence on the quantum numbers were somehow more complicated, you could for example calculate the gradient to find some minimum values, and go from there. In this case, the minimum is trivially at 1,1,1. After you have that sorted, then all you can really do is calculate the energies for nearby quantum numbers. So you'd find E₁₁₁, E₁₁₂, ... , E₁₂₂, E₂₁₂, and so on. Then you just arrange the states according to their energy.