Suppose you have a yet to be determined number of vertical lines with length 50 on which you'd like to place as many circles as you can. Each circle is 10 units in diameter and its outside edge must be at least 12 units away from outside edge of every other circle. The vertical lines upon which the circles are placed can be horizontally spaced however you'd like so long as the distance between the the left-most and right-most vertical lines does not exceed 100 units and the number of lines isn't to exceed 10. Where should you place the lines and circles so that you can maximize the number of circles placed?
I thought about taking a Linear Programming approach to this problem but as you move circles along lines, the distance to other circles varies non-linearly. Also the number of decision variables would be dependent on how many circles were placed. That is, I'd like to know the coordinates of each circle placed. What sort of tools are needed to solve this problem?
Use 6 lines (20 units between lines) make circles on alternate lines, place circles in on odd lines in half the eight of the even. You did not specify the length of the lines.