Let $E$ be a set of $n$ points in the plane $(n \geq 3)$ whose coordinates are integers such that any three points from $E$ are vertices of a nondegenerate triangle whose centroid doesnt have both coordinates integers. Determine the maximal $n.$
1. **Reduction Modulo 3:**
   Consider the set \( R \) of pairs of coordinates of the points from \( E \) reduced modulo 3. This means that each point \((x, y) \in E\) is mapped to \((x \mod 3, y \mod 3)\).

2. **Nondegenerate Triangle Condition:**
   If some element of \( R \) occurs thrice, then the corresponding points are vertices of a triangle with integer barycenter. This is because the centroid of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is given by:
   \[
   \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)
   \]
   If \( x_1 \equiv x_2 \equiv x_3 \pmod{3} \) and \( y_1 \equiv y_2 \equiv y_3 \pmod{3} \), then \( x_1 + x_2 + x_3 \equiv 0 \pmod{3} \) and \( y_1 + y_2 + y_3 \equiv 0 \pmod{3} \), making the centroid have integer coordinates.

3. **Distinct Coordinates Condition:**
   No three elements from \( E \) can have distinct \( x \)-coordinates and distinct \( y \)-coordinates. This is because if they did, their centroid would have integer coordinates, violating the given condition.

4. **Bounding the Size of \( R \):**
   By an easy discussion, we can conclude that the set \( R \) contains at most four elements. This is because there are only 9 possible pairs \((x \mod 3, y \mod 3)\) and we need to avoid having three points with the same pair.

5. **Maximal Size of \( E \):**
   Since each element in \( R \) can correspond to at most two points in \( E \) (to avoid having three points with the same pair), the maximal size of \( E \) is \( 4 \times 2 = 8 \).

6. **Example of a Set \( E \):**
   An example of a set \( E \) consisting of 8 points that satisfies the required condition is:
   \[
   E = \{(0, 0), (1, 0), (0, 1), (1, 1), (3, 6), (4, 6), (3, 7), (4, 7)\}
   \]
   These points, when reduced modulo 3, give:
   \[
   \{(0, 0), (1, 0), (0, 1), (1, 1), (0, 0), (1, 0), (0, 1), (1, 1)\}
   \]
   which satisfies the condition that no three points have the same pair of coordinates modulo 3.

The final answer is \( \boxed{8} \).