There are 2008 red cards and 2008 white cards. 2008  players sit down in circular toward the inside of the circle in situation that 2 red cards and 2 white cards from each card are delivered to each person. Each person conducts the following procedure in one turn as follows.

$ (*)$ If you have more than one red card, then you will pass one red card to the left-neighbouring player.

If you have no red card, then you will pass one white card to the left -neighbouring player.
Find the maximum value of the number of turn required for the state such that all person will have one red card and one white card first.
1. **Initial Setup**: 
   - There are 2008 players sitting in a circle.
   - Each player starts with either 2 red cards or 2 white cards.
   - The total number of red cards is 2008, and the total number of white cards is 2008.

2. **Turn Mechanics**:
   - If a player has one or more red cards, they pass one red card to their left-neighboring player.
   - If a player has no red cards, they pass one white card to their left-neighboring player.

3. **Objective**:
   - We need to determine the maximum number of turns required for each player to end up with exactly one red card and one white card.

4. **Analysis**:
   - Initially, each player has either 2 red cards or 2 white cards.
   - The goal is to reach a state where each player has exactly one red card and one white card.

5. **Card Passing Dynamics**:
   - Consider the movement of red cards. Each turn, a red card is passed to the left by players who have red cards.
   - Similarly, white cards are passed to the left by players who have no red cards.

6. **Symmetry and Maximum Turns**:
   - Since the players are arranged in a circle, the problem is symmetric. The maximum number of turns will occur when the cards are distributed in the most inefficient manner.
   - The worst-case scenario is when all red cards are initially grouped together, and all white cards are grouped together.

7. **Worst-Case Scenario**:
   - Suppose all 2008 red cards are with the first 1004 players, and all 2008 white cards are with the remaining 1004 players.
   - In each turn, the boundary between red cards and white cards will shift by one position.

8. **Calculating Maximum Turns**:
   - To achieve the desired state, the boundary must shift 1004 positions (half the circle) to ensure that each player has one red card and one white card.
   - Therefore, the maximum number of turns required is 1004.

The final answer is $\boxed{1004}$.