Lamp switches in cube pieces
Source: Spain MO 2023 P1
March 10, 2023
Problem Statement
A 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.