Square covered with circles

Question

I have a square 800x800 and i need to fully cover it with the least number of circles possible, each circle has a radius of 150.

QUESTIONS: - What pattern would be the best to use? Clover, diamon or something different? - How should i dispose the circles, and how many of them are needed? - Is there a relation to the data above and the number of circles?

Answer 1

The following site has diagrams for $1$ to $12$ circles. Unfortunately, it stops when the square side is just less than five times the circle radius.

https://erich-friedman.github.io/packing/circovsqu/

Sixteen circles ranged in a regular, square pattern $200$ apart, at $(100,100),(100,300)$ and so on will cover the $800$ square.

Answer 2

This is not an answer but some partial findings.

As shown in the figure at end, 15 circles are enough to cover the square.

To construct the covering, you first place 4 circles centered on the diagonal. These are the 4 red circles in the figure. Let look the red circle centered at $A$ in the figure. Let's say its boundary intersect an edge of the square at $E$. Let $F$ be the mid-point on that edge. Construct a circle centered at $E$ with radius 300 and let it intersect the perpendicular bisector of the edge at $R$. If one construct a circle passing through $E, F$ and $R$. You obtain one of the green circles in the figure. Replicate it by symmetry, you get 8 green circles. Together with the 4 red circles, these 12 circles covered the edges of the square and left with a hole in the middle. From the figure, one can see that one can cover the hole with 3 circles.

Notice the hole is nearly filled by a single circle. It might be possible to reduce the number by 1 but I can't figure out how to do that.

Answer 3

For the following arrangement of $11$ points, the minimum distance (indicated by the blue line segment) between pairs is $>300$, implying a lower bound of $11$ circles:

I obtained the following approximate solution with $14$ circles by randomly generating $10000$ points and solving the corresponding set cover problem. There are still some gaps, but maybe it can be tweaked to cover them:

Answer 4

Based on my code written for this answer on puzzling.se I found a solution with 14 circles:

Coordinates:

 0, 278.94802, 266.51974
 1, 494.65958, 668.05464
 2, 278.26437,   2.45011
 3, 681.45341, 711.12797
 4, 494.75679, 131.85939
 5, 701.33277, 289.36821
 6,  65.60149, 400.01636
 7, 487.09331, 399.91577
 8, 278.98163, 533.39755
 9,  64.97167, 666.56928
10,  64.93806, 133.44708
11, 701.27443, 511.03548
12, 681.50238,  89.15878
13, 278.33274, 797.62037