Recently, I've been having a lot of discussions with colleagues about equivalence resistance, and the difference between teaching the concept as a mathematical tool for thinking (and problem-solving) vs. teaching the parallel and series equations in order to calculate resistances. So far, not a single colleague has demonstrated much facility with mathematical thinking about equivalent resistance situations without resorting to the equations. Once I show them how, however, they are off to the races, and often pretty excited about their new mathematical tool.
Here are two problems that are fun if you think about the concept and simply miserable if you approach it using guess-and-check-with-equation.
Using only 10-Ohm resistors, come up with 4 different ways of hooking them up to a single (ideal) battery such that the equivalent resistance across the battery is 10 Ohms.
Using only 3-Ohm and 2-Ohm resistors, come up with 4 ways of making a 1-Ohm equivalent resistor circuit element.