3. Below you see a square with side 1. From the two top corners of that square, two spotlights shine on the bottom side. The left spotlight illuminates exactly two-thirds of the bottom side to the right, and the right spotlight illuminates exactly two-thirds of the bottom side to the left. In the figure, you can see that there is a part (in dark gray) that is illuminated by both spotlights.

What is the area of the dark gray region?

![](https://cdn.mathpix.com/cropped/2024_04_17_56b78c3c74539692475ag-3.jpg?height=317&width=325&top_left_y=1012&top_left_x=1522)
B3. The dark gray area, plus the two light gray triangles at the bottom left and right, together form a triangle that is exactly $\frac{1}{4}$ of the square. To determine the area of the dark gray part, we need the area of these small light gray triangles. The base of such a triangle is $\frac{1}{3}$.

For the height, we look at the quarter square in the bottom left of the picture, see the figure next to it. There you see two congruent light gray triangles: their height is thus half the height of the figure, which is a quarter of the original square.

![](https://cdn.mathpix.com/cropped/2024_04_17_a4374d618d1fd957f1eeg-4.jpg?height=163&width=171&top_left_y=1352&top_left_x=1619)

The area of a light gray triangle is therefore $\frac{1}{2} \cdot \frac{1}{3} \cdot \frac{1}{4}=\frac{1}{24}$. From this, it follows that the area of the dark gray area is $\frac{1}{4}-2 \cdot \frac{1}{24}=\frac{1}{6}$.