Problem 1188

To solve this problem, we need to understand the sequence of operations and how they affect the position of the cards. Let's denote the initial position of the cards as $a_1, a_2, \ldots, a_{2008}$, where $a_i$ represents the card with number $i$.

1. Understanding the Operations:
- Step (i): The top card is placed at the bottom.
- Step (ii): The new top card is removed.

2. Simulating the Process:
- Initially, the cards are in the order $1, 2, 3, \ldots, 2008$.
- After the first move, the card $1$ is placed at the bottom, and card $2$ is removed.
- After the second move, the card $3$ is placed at the bottom, and card $4$ is removed.
- This process continues until all cards are removed.

3. Pattern Recognition:
- We need to determine the initial position of the card that ends up being removed last, which is card $2008$.
- Notice that the sequence of removals follows a specific pattern. The card that is removed in the $n$-th position is the card that was initially in the $2n-1$-th position.

4. General Formula:
- For $n = 2^m k$ where $k$ is odd, the card that ends up in the $n$-th position is given by $n - \frac{k-1}{2}$.

5. Applying the Formula:
- For $n = 2008$, we need to express $2008$ in the form $2^m k$.
- $2008 = 2^3 \cdot 251$, where $251$ is odd.
- Using the formula, the initial position of the card that ends up being removed last is:
$$2008 - \frac{251-1}{2} = 2008 - 125 = 1883$$

Therefore, the card that was at the top before the game started is card number $1883$.

The final answer is $\boxed{1883}$.