Looking for solution to “lights out” puzzle variant with multiple states

Recently in World of Warcraft, there is a puzzle that is very similar to the "lights out" puzzle where a player needs to flip switches to turn all the lights into a specific color (in this case yellow, green, red, white). I have seen other solution to the lights out problem using linear algebra however all these uses only 2 states (on or off).

I haven't ever ran into a system of linear equation with modular operation before and would like some help solving something like:

  L_1 = ((s_1 + s_2 + ...s_n + c_1 ) mod 4)

L_2 = ((s_1 + s_2 + ...s_n + c_2 ) mod 4)

...

L_n = ((s_1 + s_2 + ...s_n + c_n ) mod 4)


where each L has some linear combination of s + constant mod 4

• It will be a bit different mod 4 because 2 doesn't have a modular inverse, so mod 4 doesn't form a field. But you can still do gaussian elimination on the matrix, just do all the adding and mulitplying mod 4 and try hard to pick odd numbers to make into pivots. – DanielV Dec 17 '18 at 1:27
• See the 2017 article "Lights Out" and Variants by Martin Kreh in the American Mathematical Monthly which specifically treats modular colors on square grids. – Brian Hopkins Dec 17 '18 at 3:21

First treat states $$1$$ and $$3$$ as if they are switched-on lights, and states $$0$$ and $$2$$ as if they are switched-off lights. Solve this as the normal lights out. You are essentially just working mod $$2$$, making everything even. At the end of this stage, all lights are either $$0$$ or $$2$$.
Then solve the rest as another two-state lights out puzzle, where $$2$$ is the switched-on state and $$0$$ is a switched-off state. The only difference is that the moves you do consist of double button presses. A double button press skips over the $$1$$ and $$3$$ states, and toggles lights between the $$0$$ and $$2$$ state.