Adjacent-corner resistance of a resistor cube ($R_{\mathrm{adj}}=7R/12$)

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.