That Manipulate has a minor problem in that a Slider is not the most appropriate method for selecting a small positive integer. A SetterBar might be better. It has a major problem in that it fails to provide a fixed background. Every dynamic presentation requires a fixed background. In this case the size of the circles and display should be fixed but here the entire display changes when there is only one disk. Another minor problem is that Orange characters on a Green background is not as readable as we might wish.
The following is a custom dynamic display constructed with the Presentations Application. It contains a number of "convenience routines" that I believe make it fairly easy to construct custom dynamic displays.
First we write a specification for the labeled disks. In this case it is convenient to define positions as numbers in the complex plane. So ComplexText centers text at the complex location z. ComplexCirclePoint draws an outlined colored disk of a given point size at z. I decided to go with Black characters on regular Green disks.
The following is the specification for the dynamic presentation, which I hope is rather intuitive. The constructions pagelet and phrase are basically shortcuts for Column and Row constructions without the braces. The second argument in Dynamic allows us to calculate the angle in the complex plane each time the number of disks is reset. I decided to allow zero disks as one choice. Draw2D is basically the same as Graphics but with its own set of Options. The background for the display is invariant and the first disk, when present, always appears in the same place.
I am looking for a way that I might be able to make the disks (or any object for that matter) be arranged in a circular or pentagonal arrangement as opposed to rows. I tried looking at the "Sum of Empty Skies for a Set of Planets" to see how the "planets" were arranged, but I could not figure out how the code worked. I am thinking that there might be a function that I have not considered, and would appreciate if someone could point me in the right direction.