I’m not sure where to start.
I think the problem (finding the resistance between two particular points) would be just as hard to solve regardless of the points we choose. But I’m not sure. The system is linear, but right now I don’t see any way to use superposition or anything.
Any takers?
This problem is presented in Purcell’s Electricity and Magnetism, volume 2 of the Berkeley Physics Course, in the problems at the end of one section. They ask the reader the effective resistance between adjacent nodes. They mention that the resistance between a couple of other nodes (not sure if they’re the ones pictured) was something like pi ohms. There’s a reference to the article where that problem is solved . I don’t have my copy of Purcell here, or I’d give the reference to you.
On the XKCD forums, somebody said that you solve this with a 2D Fourier series. Apparently the answer is:
4/pi-.5 Ohms
I’ve been meaning to tackle this one, but haven’t had a chance, yet. I do know that there’s a clever trick you can use for two adjacent nodes, which doesn’t require any infinite series or calculus at all (the answer is 1/2 ohm), and I want to see if there’s some way of extending that trick for the general case (I’m almost certain there is).
I have no answer, not being a particularly math-y type of geek, but this thread answers my question about whether a math-y geek would feel compelled to try to solve it after seeing it in the strip. 
That’s given in Rysto’s link, in the paragraph below Eq. (37). It doesn’t seem to be extendable beyond adjacent nodes, however.
I assume that the problem is supposed to neglect changes as the current propagates across an infinite grid at the speed of light, right?
(Not that I have the math know how to solve it anyway…)
This problem was used in one of those quizzes that Google sends to people.
I’m pretty sure now I could google the answer, but I’m looking for some insight into how to go about solving it. I’d like to crack the armor so to speak. The link above gives a pretty nasty integral that apparently solves the general case (i.e., put a ground at node (0,0), inject one amp into node (x,y), what is the voltage across the nodes?). I don’t really know the motivation for using a Fourier transform, and instead of reading more into the derivation, I decided to step back and give it a think for awhile.
I’m off to attempt a few special cases (adjacent/diagonal nodes) before trying again to crack the general case.
Just wanted to say I knew this was XKCD before I even clicked the OP link… 
Yeah, same here.
(BTW, the problem is far beyond my abilities, though I did guess that an infinite series would be involved.)