OME 2023 Problem 1

A $3\times 3\times 3$ cube is made of 27 unit cube pieces. Each piece contains a lamp, which can be on or off. Every time a piece is pressed (the center piece cannot be pressed), the state of that piece and the pieces that share a face with it changes. Initially all lamps are off. Determine which of the following states are achievable: (1) All lamps are on. (2) All lamps are on except the central one. (3) Only the central lamp is on.