We have n bulbs (B_1,B_2,...,B_n) and n buttons(b_1,b_2,...,b_n) connected to each other with a box. Whenever we switch a button, the condition of the bulb which is connected to the button changes;that is, if it was ON before, now it is OFF, or vice versa. We know that each b_i button - (i=1,...,n) - is connected to the B_i bulb, but additionally it can be connected to other bulbs. Besides, if b_i is connected to B_k, then b_k is connected to B_i. So if all bulbs are OFF at the beginning, prove that there exists an order in which they become ON at the same time. Okay, i have some ideas about this but don't know how to use the information "Besides, if b_i is connected to B_k, then b_k is connected to B_i." Is it somehow related to parity?