Or, how to make 2+3=6.
Time to come close to home.
My car has one of these. It’s by Getrag (now Magna GT) and is a six speed automated manual gearbox. The picture’s shamelessly stolen from a guy in Québec.
Let’s look at the first picture. You’ve got the final drive and differential with two gears (top left), a cylinder with three wavy grooves in it* (bottom left), a shaft with various gears and rods (middle) and another shaft with three gears (right). The cylinder with the three grooves* engages with the three rods on the complicated looking shaft as on the second image. Rotate the cylinder and the rods move from side to side, engaging one or other of the three gears. But it’s a six speeder. Where are the other three?
That’s where the two gears on the final drive come in. They provide two ranges. You (or to be specific an electric motor) change from first to third using the lower of the two ratios in the final drive then you go back to first and simultaneously switch to the higher final drive ratio. Now first becomes fourth and you proceed upwards. Just like a heavy truck then, albeit with fewer gears.
*it has a proper name too!
So there you have it. Five ratios become six because it’s 2x3, not 2+3.