Balancing redox equations with oxidation numbers

Trial and error works for simple equations. It fails badly for things like

$\text{MnO}_4^- + \text{Fe}^{2+} + \text{H}^+ \rightarrow \text{Mn}^{2+} + \text{Fe}^{3+} + \text{H}_2\text{O}$

There are too many species to fiddle with. Oxidation numbers turn the problem into arithmetic.

The key idea: electrons lost = electrons gained. The total increase in oxidation number must equal the total decrease.

1. Assign oxidation numbers to the elements that change, on both sides.
2. Find the change per atom: an increase is electrons lost (oxidation); a decrease is electrons gained (reduction).
3. Choose coefficients so that total increase = total decrease.
4. Balance the other atoms by inspection — metals first, then the main non-metal.
5. Balance oxygen by adding $\text{H}_2\text{O}$ to the side that needs it.
6. Balance hydrogen by adding $\text{H}^+$ (in acidic conditions).
7. Check charge on both sides. This is the step that catches mistakes.

In alkaline conditions you would add $\text{OH}^-$ instead of $\text{H}^+$ at the end, but Cambridge redox balancing is usually in acid.