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I have a 9-way switch like this guy:

And I'm trying to figure out how I can turn on one LED with position 1, 2 with position 2, all the way up to all 9 in position 9.

Obviously I can repeat all the wiring for the LEDs at each position, but that seems silly.

My idea is that with a layout like below, the switch would represent the circled red line (shown in position 3), which would elongate to the right in each successive position until it connects all the lights. How can I do this?

use a regulated current source to light them, wire them in series and short out the segemnt you want to be dark.

^{simulate this circuit – Schematic created using CircuitLab}

you can possibly use a buck-boost converter to make the 30V if you don't already have a suitable voltage.

here's a simple way to build one using a LM2596S module
remove the potentiometer, and both large capacitors, connect one of the salvaged
capacitor between +in and +out (positive to +in), fit a 1uF ceramic capacitor where the output capacitor was. connect a 100 ohms resistor from the -output to the centre potentiometer terminal.

modified in this way it will create a negative voltage on the -out terminals and act as a 12.5mA current sink at the centre potentiometer terminal (with source at +out) if power is applied between +in and +out

^{simulate this circuit}

Here is a low tech solution that requires a lot of parts. Only 4 positions shown, you need 45 diodes for 9 positions.

Sunyskyguy has a clever solution if you have a high voltage available.

If you can afford another 0.5V drop you can use a massive array of diodes. Here is an example with three LEDS which requires 6 diodes.

(Sorry for the SW, SW2.., circuit lab does not have a rotatory switch symbol)

^{simulate this circuit – Schematic created using CircuitLab}

If you’re not wedded to the specific switch you have, get a “progressive shorting rotary switch” to replace it. That works just like your drawing.

It might seem like overkill but it would be fewer parts, and possibly less expensive, than some other solutions to use a micro-controller. Many Ardunio boards have over 9 digital output pins - you could drive one LED with each of nine pins. By having the switch pick different points of a voltage divider and feeding it to one analog pin, you could determine the switch position and light up what ever you have decided should be lit up.