From one of my favorite books, Code: The Hidden Language of Computer Hardware and Software by Charles Petzold:

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Here, he explains how to setup switches and lightbulbs in two locations with the purpose of communicating by turning the lights on and off according to some previously established protocol/code.

My question: is he oversimplifying - because that isn’t the point of this chapter - which route/path the current would take when both switches are closed? As in:

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Would the wire, which I awkwardly marked blue, also become charged and/or carry current? Or would there be no current because the cathodes of the two adjacent batteries cancel out any difference/voltage that would otherwise cause a current?