Find largest value of $n$ for condition to hold true Given the unit circle $\mathbb{U}=\left\{(x,y)\vert x,y\in\mathbb{R},0\leq x^2+y^2\leq1\right\}$, and a little bigger circle $\mathbb{U^*}=\left\{(x,y)\vert x,y\in\mathbb{R},0\leq x^2+y^2\leq1+k\right\}$,$\mathbb{U}\subset\mathbb{U^*}$, and a region
$$P_i=P(x_i,y_i)=\left\{(x,y)\in\mathbb{U^*}\vert(x_i,y_i)\in\mathbb{U},(x-x_i)^2+(y-y_i)^2\leq k^2\right\}$$
for $0<k\le\frac{1}{2}$
and
$$d(P_i,P_j)=\sqrt{(x_i-x_j)^2+(y_i-y_j)^2}$$
Find the largest value of $n$ in function of $k$ if
$$\left(\bigcup_{i=1}^{n}P_i\right)\cap P_{n+1}\neq\emptyset\tag 1$$
$$\left(\bigcup_{i=1}^{n-1}P_i\right)\cap P_{n}=\emptyset\tag 2$$
must both be true.

From (2):
$$(P_1\cap P_n)\cup\dots\cup(P_{n-1}\cap P_n)=\emptyset$$
That means
$$P_1\cap P_n=\emptyset\rightarrow d(P_1,P_n)\geq2k$$
$$\vdots$$
$$P_{n-1}\cap P_n=\emptyset\rightarrow d(P_{n-1},P_n)\geq2k$$
but I don't know how to follow from there.
 A: I'd view this the other way round: given a number $n$ of disks, how large can they be, i.e. what's the maximal $k$? See circles in circles in Erich's Packing Center which lists $\frac{k}{1+k}$, i.e. radius of individual disks $P_i$ as a fraction of the circle containing all these disks, your $\mathbb U^*$.
The arrangement using $n=12$ disks is listed without proof. So although there is some evidence suggesting that the listed arrangement and the radius it uses is indeed optimal, we don't know that for sure. Or at least the author of that page didn't know that at the time of writing. Even if the case of $n=12$ were solved by now, I doubt someone has come up with a general formula which works in all such situations.
So you can't expect to have a simple formula here. You can hope to find some publications listing the fractions for several explicit values of $n$, some with proof of optimality and many without. If that is not enough, you will have to go for approximate solutions, without expecting optimality.
