Problem 1241

1. Identify the midpoints and draw the segments:
- Let the square be $ABCD$ with side length $s$.
- Mark the midpoints $M$ and $N$ of sides $AB$ and $BC$ respectively.
- Draw segments $AM$, $BN$, $CM$, and $DN$.

2. Analyze the quadrilaterals formed:
- The segments divide the square into four quadrilaterals.
- The quadrilateral with more dense cross-hatching is formed by the segments $AM$ and $DN$.
- The quadrilateral with less dense cross-hatching is formed by the segments $BN$ and $CM$.
- The middle quadrilateral is formed by the intersection of all four segments.

3. Calculate the areas of the quadrilaterals:
- The area of the entire square is $s^2$.
- Each of the four quadrilaterals is a right triangle with legs of length $\frac{s}{2}$.
- The area of each right triangle is $\frac{1}{2} \times \frac{s}{2} \times \frac{s}{2} = \frac{s^2}{8}$.

4. Combine the areas:
- The middle quadrilateral is formed by the intersection of the four right triangles.
- The area of the middle quadrilateral is $\frac{s^2}{4}$ (since it is half of the area of the square minus the areas of the four right triangles).

5. Compare the areas:
- The quadrilateral with more dense cross-hatching is formed by combining two right triangles.
- The area of this quadrilateral is $2 \times \frac{s^2}{8} = \frac{s^2}{4}$.
- The quadrilateral with less dense cross-hatching is also formed by combining two right triangles.
- The area of this quadrilateral is also $\frac{s^2}{4}$.

6. Conclusion:
- Since the middle quadrilateral has an area of $\frac{s^2}{4}$ and the quadrilateral with more dense cross-hatching has an area of $\frac{s^2}{4}$, the quadrilateral with more dense cross-hatching has the largest area.

The final answer is the quadrilateral with more dense cross-hatching.