Tracey baked a square cake whose surface is dissected in a $ 10 \times 10$ grid. In some of the fields she wants to put a strawberry such that for each four fields that compose a rectangle whose edges run in parallel to the edges of the cake boundary there is at least one strawberry. What is the minimum number of required strawberries?
To solve this problem, we need to ensure that every possible rectangle within the $10 \times 10$ grid contains at least one strawberry. We will use a combinatorial approach to determine the minimum number of strawberries required.

1. **Understanding the Problem:**
   - We have a $10 \times 10$ grid, which means there are $100$ fields.
   - We need to place strawberries such that every rectangle formed by the grid lines contains at least one strawberry.

2. **Formulating the Problem:**
   - Consider a rectangle within the grid. The vertices of the rectangle are intersections of the grid lines.
   - For a rectangle to be valid, it must have at least one strawberry in its interior.

3. **Strategy:**
   - We can use a combinatorial argument to place strawberries in such a way that every possible rectangle is covered.
   - One effective strategy is to place strawberries in a checkerboard pattern.

4. **Checkerboard Pattern:**
   - In a checkerboard pattern, we place strawberries in alternating cells.
   - This ensures that any rectangle will have at least one strawberry because the pattern guarantees coverage of all possible rectangles.

5. **Calculating the Minimum Number of Strawberries:**
   - In a $10 \times 10$ grid, a checkerboard pattern will place strawberries in half of the cells.
   - Therefore, the number of strawberries required is $\frac{10 \times 10}{2} = 50$.

6. **Verification:**
   - To verify, consider any rectangle within the grid. Since the strawberries are placed in a checkerboard pattern, any rectangle will intersect with at least one cell containing a strawberry.

Thus, the minimum number of strawberries required to ensure that every rectangle contains at least one strawberry is $50$.

The final answer is $\boxed{50}$.