MathDB

1

Part of 2023 Spain Mathematical Olympiad

Problems(1)

Lamp switches in cube pieces

Source: Spain MO 2023 P1

3/10/2023
A 3×3×33\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.