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Expected number of moves needed to get to certain position

Suppose we have a balanced $n \times n$ board with $1$’s and $0$’s and we move a single $1$ through the board without flipping any $0$’s. For example, the board below is balanced.
Is it true that for any position of the board there is a unique path to it through the board? If so, how many moves are needed to reach a position from an arbitrary starting position?

A:

If you only move a single $1$, the number of positions you can reach in $k$ moves is $2^k$. To see this, start with a single $1$, and then place the first $1$ in one of the $k$ cells on the left side of the board. This is clearly a valid path. Now, just mirror the rest of the board.

A:

This is an interesting problem.
For simplicity, suppose that the $n \times n$ board has only two colours: red and blue, and we want to move a blue unit from the origin to the point $(i,j)$. We need to prove that there is a unique sequence of moves to get to $(i,j)$ starting from the origin and using only red moves.
Let $P$ be the sequence of moves we need. By definition of “red move” we know that $P$ is either:

a sequence of red moves, or
a sequence of blue moves followed by a sequence of red moves.

Notice that if the sequence of moves $P$ starts with a blue move, then the sequence $P’$ where we reverse the order is also a valid sequence. Notice also that if $P’$ begins with a sequence of red moves, then this sequence can be reverse (this is because a blue move must always be followed by a red move). Therefore, we only have to consider the case where $P$ is a sequence of red moves.
Let $D$ be the maximum value of $|P|$. Notice that $D$ cannot be more than $n$ because we can get

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