This link has a lot of information on this and related problems. It gives \(R_{eq}=R/2\) in this case, and proves it first in a simple (albeit non-rigorous) way, and then provides rigorous justification. It also gives equivalent resistances for other points \(A\) and \(B\).

" Thus the direct link carries the same current as all the other paths, so the resistance of the direct link equals the effective resistance of the entire grid excluding that link. The direct link is in parallel with the remainder of the grid, so the combined resistance is simply R/2. " - why the resistance of the direct link is equally to the effective resistance of the entire grid EXCLUDING that link?

Let potential difference V be applied across A and B and i be the current in wires.
Assume A is isolated positive terminal, so in every direction it sends out a current of i/4 (as there are 4 directions current at a junction can take). Similarly assume B is an isolated negative terminal, so from every direction it recieves an incoming current of i/4. Therefore, in the wire between A and B, net current is i/2 from A to B and hence resistance between A and B is R(0)/2

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thnks a lot! :)

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This link has a lot of information on this and related problems. It gives \(R_{eq}=R/2\) in this case, and proves it first in a simple (albeit non-rigorous) way, and then provides rigorous justification. It also gives equivalent resistances for other points \(A\) and \(B\).

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" Thus the direct link carries the same current as all the other paths, so the resistance of the direct link equals the effective resistance of the entire grid excluding that link. The direct link is in parallel with the remainder of the grid, so the combined resistance is simply R/2. " - why the resistance of the direct link is equally to the effective resistance of the entire grid EXCLUDING that link?

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Let potential difference V be applied across A and B and i be the current in wires. Assume A is isolated positive terminal, so in every direction it sends out a current of i/4 (as there are 4 directions current at a junction can take). Similarly assume B is an isolated negative terminal, so from every direction it recieves an incoming current of i/4. Therefore, in the wire between A and B, net current is i/2 from A to B and hence resistance between A and B is R(0)/2

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irodov right??

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Yes

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A Twist : What if these points A and B were diagonally opposite??

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