Yatta and Yogi play a game in which they begin with a pile of $n$ stones.  The players take turns removing $1$, $2$, $3$, $5$, $6$, $7$, or $8$ stones from the pile.  That is, when it is a player's turn to remove stones, that player may remove from $1$ to $8$ stones, but [i]cannot[/i] remove exactly $4$ stones.  The player who removes the last stone [i]loses[/i].  Yogi goes first and finds that he has a winning position, meaning that so long as he plays perfectly, Yatta cannot defeat him.  For how many positive integers $n$ from $100$ to $2008$ inclusive is this the case?
1. **Understanding the Game Rules**:
   - Players can remove $1, 2, 3, 5, 6, 7,$ or $8$ stones.
   - The player who removes the last stone loses.
   - Yogi goes first and has a winning strategy.

2. **Analyzing Winning and Losing Positions**:
   - A losing position is one where any move leaves the opponent in a winning position.
   - A winning position is one where there exists at least one move that leaves the opponent in a losing position.

3. **Identifying Key Positions**:
   - If Yogi can force Yatta into a position where the number of stones is $9k + 1$ (for some integer $k$), Yatta will be in a losing position.
   - This is because Yatta cannot remove $4$ stones, and any other move will leave Yogi with a number of stones that is not $9k + 1$.

4. **Mathematical Formulation**:
   - We need to count the number of integers $n$ from $100$ to $2008$ that are of the form $9k + 1$.
   - These are the positions where Yogi can force Yatta into a losing position.

5. **Counting the Relevant Numbers**:
   - The sequence of numbers of the form $9k + 1$ within the range $100$ to $2008$ can be found by solving:
     \[
     100 \leq 9k + 1 \leq 2008
     \]
   - Subtract $1$ from all parts of the inequality:
     \[
     99 \leq 9k \leq 2007
     \]
   - Divide by $9$:
     \[
     11 \leq k \leq 223
     \]
   - Therefore, $k$ ranges from $11$ to $223$ inclusive.

6. **Calculating the Number of Values**:
   - The number of integers $k$ from $11$ to $223$ is:
     \[
     223 - 11 + 1 = 213
     \]

The final answer is $\boxed{213}$.