OMM 2020 Problem 4

Let $n\ge 3$ be an integer. In a game there are $n$ boxes in a circular array. At the beginning, each box contains an object which can be rock, paper or scissors, in such a way that there are no two adjacent boxes with the same object, and each object appears in at least one box.

Same as in the game, rock beats scissors, scissors beat paper, and paper beats rock.

The game consists on moving objects from one box to another according to the following rule:

Two adjacent boxes and one object from each one are chosen in such a way that these are different, and we move the loser object to the box containing the winner object. For example, if we picked rock from box A and scissors from box B, we move scossors to box A.

Prove that, applying the rule enough times, it is possible to move all the objects to the same box.

Proposed by Victor de la Fuente