Lamppost Toggle Cycle on a Square

Four lampposts sit at the corners of a square, labeled $A, B, C, D$ in clockwise order. All four lamps start on. A walker moves clockwise around the square, visiting one lamppost per step. The toggle rule is: - When the walker arrives at a lamppost, if the lamp at the **previous** lamppost is currently **on**, toggle the current lamp (flip its state). - If the previous lamp is **off**, do nothing to the current lamp. The walker starts at lamp $A$ (before taking any steps). (a) Model this system as a finite deterministic dynamical system. Describe the state space (lamp states plus walker position). (b) Starting from all lamps on with the walker at $A$, compute the lamp configuration after $N$ steps as an explicit function of $N$. Give the configuration for each step in the first period. (c) Prove the process is eventually periodic and find the minimal period.