For a given shape (e.g. square) and a given number (e.g. 15) of non-overlapping unit circles in the shape, there is an optimal arrangement of the circles that minimizes the area of the shape. (examples)
Is every optimal arrangement unique? Or is there some shape and some number of circles such that the minimum area of the shape is achieved by different arrangements?
If two arrangements (just considering the circles, not the shape) are mirror images or rotations of each other, I consider them to be the same. Rattlers (circles that can move without causing other circles to move) are not considered when determining if two arrangements are the same.
Try these two non-equivalent optimal packings of $4$ circles in an L-shaped region.
You can put in small indentations to prevent "rattlers" from rattling, or instead of the L take the "shape" to be the union of the circles.
EDIT: Here's an example where the shape is convex.
