Consider a row of 3 lights, each of which can be in one of the 3 states: off, light blue, and dark blue. The "next state" of each state is by the following cyclic order: off → light blue → dark blue → off. Below the three lights are 3 switches A, B, and C, each changes the state of particular lights to the “next state”: Switch A changes the states of the first two lights, switch B changes the states of all three lights, and switch C changes the states of the last two lights. If initially all three lights are off, can you push the switches in some order, such that the three lights are of the following states: off, light blue, and dark blue in a row? (Hint: Similar to watermellon problem, first write down a linear system to represent the effect by pushing A for x times, B for y times, and C for z times. But you need to work on the integer number system modulo 3, i.e. x, y, z can only take values in 0, 1, 2 mod(3), to code the cyclic ordered 3 states).