An intriguing one -II

This is the continuation of the post 'An intriguing one'. Here I make an attempt to look at a possible approach.

Consider two concentric circles with radii and . It is clear that there cannot exist two points within the b-circle.

I go on placing points on the circumference of a-circle with gap of at least 'b' of course, I do not know what to do with . As of now we shall distribute evenly. This was to give an idea. You must have figured out that we can pack still more, it is not difficult to see that we can place
points. So the picture now looks like fig1.


I have written only two circles with centers and respectively, to show available region for further points. After doing this, pick points like and in the next iteration. But analysis becomes non trivial from this step itself.

Of course the problem is solved of if points like lie within b-circle.

Miscellany: Seemingly related but a easy one i found at Colorado mathematical Olympiad

(a) We need to protect from the rain a cake that is in the shape of an equilateral triangle of side 2.1. All we have are identical tiles in the shape of an equilateral triangle of side 1. Find the smallest number of tiles needed.
(b) Suppose the cake is in the shape of an equilateral triangle of side 3.1. Will 11 tiles be enough to protect it from the rain?

I found that we require 6 and 11 .