1. Given that $n$ is a positive integer, Tom and Jerry are playing a game on a board with 2018 squares in a row. At the beginning, Jerry chooses one square to place a coin. In each subsequent step, Tom says a positive integer between $1$ and $n$ (inclusive of $1$ and $n$), and Jerry decides whether to move the coin left or right by the number of squares specified. When Jerry can no longer move, Tom wins. Find the smallest $n$ such that Tom can win in a finite number of steps by choosing appropriate numbers.
1. First prove the smallest $n$ is 1010.

If $n \leqslant 1009$, then Jerry has a simple strategy to ensure the coin can move from any cell, meaning he will never lose immediately.

Assume the coin is in a cell where it cannot move further, then there are at most $n-1$ cells to the left of this cell, and at most $n-1$ cells to the right, so the board has at most
$$
2(n-1)+1=2 n-1 \leqslant 2017
$$

cells, which is a contradiction.
Now assume $n=1010$. Label the cells on the board from left to right as $1,2, \cdots, 2018$, and use $k$ to denote the number of the current cell.

When $k \in\{1009,1010\}$, Tom can immediately say 1010 to ensure his victory.
If $k>1010$, Tom can use the number $k-1010$.