Kirchhoff's rules

Junction rule: the total current flowing into any junction equals the total current flowing out of it. It follows from conservation of charge in steady state.

Loop rule: going around any closed loop, the changes in potential add up algebraically to zero. It follows from the electrostatic force being conservative (energy conservation).

Assign a current with an assumed direction to each branch; if the solution gives a negative value, the actual current flows the other way, and the magnitude is still correct.

Going through a resistor in the direction of the assumed current, the potential falls by IR; going against it, it rises by IR.

Going through a cell from its negative to positive terminal, the potential rises by ε; from positive to negative, it falls by ε.

Use the junction rule first to reduce the number of unknown currents, then write as many independent loop equations as are needed.

Symmetry in a network (equal potentials at symmetric points) can remove resistors that carry no current and simplify the work.