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

  • $\begingroup$ 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. $\endgroup$
    – DanielV
    Dec 17 '18 at 1:27
  • $\begingroup$ See the 2017 article "Lights Out" and Variants by Martin Kreh in the American Mathematical Monthly which specifically treats modular colors on square grids. $\endgroup$ Dec 17 '18 at 3:21

You can solve the mod 4 version like two instances of the mod 2 version.

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.


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