MathNet 2016-part-01 — 184 problems (EN–VI) Question 71

In a computer game a $4 \times 4 \times 4$ cube is built using $4^3$ unit cubes. At the beginning of the game each unit cube contains an integer. In each turn of the game, you choose a unit cube and increase by 1 all the integers in the cubes having a face in common with the chosen cube. You win the game if you reach a position in a finite number of turns where all the $4^3$ integers are divisible by 3.
Is it possible to win the game no matter what the starting position is?