I had assumed that since Google is compute-oriented (and the alt text said it was from Google), that the solution would have been something like:
notice that you can upper bound the resistance by replacing some resistors with perfect resistors
notice that you can lower bound the resistance by replacing some resistors with perfect conductors
notice that the convergence properties of doing the above outside some finite region are pretty decent as the region expands
write a computer program to numerically compute the answer to a desired level of precision using the above
In fact, I still kind of expect that this form of answer is likely more what they were looking for, do they really expect this advanced-looking math? But, interesting to see it does have an analytical solution.
Yeah, I don’t believe it would be possible to come up with this (or the solutions on MathPages that I forgot to link, thank you Shankar) ex nihilo in the context of a job interview, unless one happened to be Ramanujan himself. If I saw something like this in an interview I would assume that they were looking for how the interviewee deals with an intractable problem. That said, it looks like the Google test that this was a part of was a take-home thing, so you could just look up the answer.
I had assumed that since Google is compute-oriented (and the alt text said it was from Google), that the solution would have been something like:
notice that you can upper bound the resistance by replacing some resistors with perfect resistors
notice that you can lower bound the resistance by replacing some resistors with perfect conductors
notice that the convergence properties of doing the above outside some finite region are pretty decent as the region expands
write a computer program to numerically compute the answer to a desired level of precision using the above
In fact, I still kind of expect that this form of answer is likely more what they were looking for, do they really expect this advanced-looking math? But, interesting to see it does have an analytical solution.
I put this to Claude to fill out the details for the numerical approach: https://claude.ai/share/d8fbe3c6-39f0-481e-b05c-d11904f23e91. Convergence not quite as good as I intuitively expected, but still works.
Yeah, I don’t believe it would be possible to come up with this (or the solutions on MathPages that I forgot to link, thank you Shankar) ex nihilo in the context of a job interview, unless one happened to be Ramanujan himself. If I saw something like this in an interview I would assume that they were looking for how the interviewee deals with an intractable problem. That said, it looks like the Google test that this was a part of was a take-home thing, so you could just look up the answer.
(I don’t know that I would be able to do this even numerically without having happened to have recently read a paper which mentioned the effective resistance)