CONO-SUR 1997 Problem 4

Consider a board with $n$ rows and $4$ columns. In the first line are written $4$ zeros (one in each house). Next, each line is then obtained from the previous line by performing the following operation: one of the houses, (that you can choose), is maintained as in the previous line; the other three are changed:

• if in the previous line there was a $0$ , then in the down square $1$ is placed;
• if in the previous line there was a $1$ , then in the down square $2$ is placed;
• if in the previous line there was a $2$ , then in the down square $0$ is placed; Build the largest possible board with all its distinct lines and demonstrate that it is impossible to build a larger board.