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Puzzle #1 (posted 04-2009)

Consider an infinite grid of 1-Ω resistors. Let's call the dimension of the grid n, where a 1-dimensional grid would be a line of resistors connected end-to-end, a 2-dimensional grid would be a rectangular array of resistors, where each resistor is connected at each end to three other resistors, a 3-dimensional grid would be a cubic lattice where each resistor is connected at end to five resistors, and so on.

What is the resistance that you would measure across a resistor, as a function on n? For the n = 1, it is just 1 Ω, but for higher n, the grid gives a parallel component that reduces the resistance that you would measure.

Hint: superposition


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