Q:

What angle does the sun have to shine for shade balls to shade 100%

Shade balls are plastic spheres that can be placed on the surface of a water reservoir to reduce the amount of sunlight hitting the reservoir. 
These spheres when packed tightly can cover almost 91% of the surface of the reservoir: 
https://youtu.be/BZbChKzedEk?t=242
The math checks out: 
$$ \frac{2 \sqrt3}{\pi} \approx 0.9069 $$
But that assumes that the sun is directly overhead. If the sun is lower in the sky, presumably it gets closer to 100%. 
What is the smallest angle of incidence that the sun has to shine where a single layer of tightly packed shade balls can shade 100% of the water's surface?

A:

In an idealised situation, the balls are arranged as in the image below (view from above). If the Sun shines from "due south", the blue ball's shadow must cover the gap north of its northern neighbours as indicated - because no other ball can as easily.
The read triangle shows the situation from the side (i.e., the hypotenuse is the water line, the long leg is the shadow boundary).
We readily read off the length of the hypotenuse as $2\sqrt 3-1$ times the ball radius. Hence the angle at the top vertex (which corresponds to the elevation angle of the point-sized Sun) is
$$\arcsin\frac1{2\sqrt 3-1} \approx 23.9^\circ$$

However, I am not yet totally sure about other Sun directions (or lattice rotations) - thus if the reference were to a differnet Youtube channel, the above image might need to be called a Parker shadow ...

