I dont know how to do it mathematically but the word retracing caught my eyes

i retraced the sequence

and obtained the following figure

 

let all lamps are lit

steps(each cycle)                                no. of lamps

 (1 on)(0 of)    

                                                                         1111(initially all lamps are lit)

(steps are retraced              )                         1000

                                                                         1100

                                                                         1010

                                                                         1111

 

after every one in previous step of previous column change the state

11

  0

as you can see

that for 4 lamps

after 4 steps back lamps are relit

proceed same way you get for power of 2

 

 

I dont know how to write it mathematically

but can these tupe of question come in IIT