Green's function and the resistance across a Hypercube

[Hamsi]

Homework Statement

I do know how to solve the resistance network problem in two dimensions. However, in this problem they want it in 3 dimensions and higher and I don't know how to do that

Relevant Equations

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Homework Statement: I do know how to solve the resistance network problem in two dimensions. However, in this problem they want it in 3 dimensions and higher and I don't know how to do that
Homework Equations: -

In the picture you can see the solution to the two dimensional version

 

[haruspex]

Green's function??

Anyway, the hint is very useful.
Start by seeing how that works in 2D. We have a square, ABCD, say. From one corner, A, there are two adjacent links, to B and D. From an electrical perspective AB and AD are parallel.
At the next step there are another two, then we reach the target, C.
The trick is that you do not need to worry about how the first pair connect to the second. Suppose we were to connect B and D as well. By symmetry, no current would flow in BD.
Now try the same method in 3D.

 

[Hamsi]

Thank you for your response.

I think I understand the problem in 2D and now I think I also understand the problem in 3D. For 3D, I got th answer $$ (5/6 )R_0$$.

However, I got this answer by using basic things as kirkhoff and rules for resistances in parallel or serie. I did not use greens function. I feel like I should, since the homerwork hints towords it.

Also, I have no idea how to do quenstion b and c. There they ask about the 4th deminsion and higher. Can you maybe give me hint for thos

 

[haruspex]

they ask about the 4th deminsion and higher

First, you need to find a general formula for the number of links that connect vertices at distance r from the starting corner to those at distance r+1 from it.
Write out those numbers for 1, 2 and 3 dimensions. In each case, the list of numbers will have a common factor n. If you factor that out, should see a familiar pattern.