I rent and the washing machine in my house is pretty old, probably 20+ years. Recently the timer knob has stopped working so I decided to take it apart to see if I could fix it.
Picture of the device in question. This is the back of it. The metal circle you see in the middle corresponds to where the knob is on the other side. Rotating around that metal circle (which looks a bit like a quarter-inch speaker plug) is a gear that looks about like this. Except on mine, it had a pattern of grooves, like as if you had zoomed in on an analog record.
These grooves corresponded to a system of eight metals connections. The grooves on the spinning timer would press a combination of the metal connections (can't find a picture of this), which would control what cycle the machine is on, kind of like a music box.
I tried to program it myself with the circuit diagram I found inside the machine and two insulated screwdrivers, but alas I could only figure out how to make it perform the rinse cycle. I quit after I accidentally made a spark so big it blinded me for a couple seconds and turned the tip of my screwdriver black.
TL;DR I'm probably going to spend the $100 bucks to buy the damned thing online.