= Adjacent-corner resistance of a resistor cube
{title2=$R_{\mathrm{adj}}=7R/12$}
If every edge of a cube has <electrical resistance> R, the <effective resistance> between adjacent vertices is $7R/12$. Apply a unit <voltage> between the terminals. Reflection symmetries leave two interior potentials x,z and their complements. <Kirchhoff's current law> gives $4x-z=2$, $4z-2x=1$, so $x=9/14$, $z=4/7$. The terminal current is $[1+2(1-x)]/R=12/(7R)$, proving the formula. No symmetry identifies vertices at different potentials across the source edge.
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