TITLE: Packing equal-size disks in a unit disk
QUESTION [7 upvotes]: Inspired by the delicious buns and Siu Mai in bamboo steamers I saw tonight in a food show about Cantonese Dim Sum, here is a natural question. It probably has been well studied in the literature, but I cannot find related reference.
Given a real number $r \le 1$, let $f(r)$ be the maximum number of radius-$r$ disks that can be packed into a unit disk. For example, $f(1)=1$ for $r \in (1/2, 1]$, $f(r)=2$ for $r \in (2\sqrt{3}-3, 1/2]$, etc.
Question: Is it true that $\{f(r): r \in (0, 1]\}=\mathbb{N}$?
Noam Elkies immediately pointed out that the maximum radius allowed for packing $6$ and $7$ balls are the same (both $1/3$). Now let me change the problem to just make it harder :)
Modified question: Is $\mathbb{N} \setminus \{f(r): r \in (0, 1]\}$ a finite set?

REPLY [13 votes]: No: according to the pictures in
https://en.wikipedia.org/wiki/Circle_packing_in_a_circle,
$f(r)$ is never $6$, with $f(\frac13) = 7$ but $f(\frac13 + \epsilon) = 5$.
(This result is attributed to the late R.L.Graham's solution in 1968 of
a problem in the American Math. Monthly.)
It is also known that $f(r_0) = 19$ for $r_0 = 1 / (1+\sqrt2+\sqrt6)$,
and conjectured that $f(r_0 + \epsilon) = 17$, which would mean that
$f(r)$ is never 18 either.   The conjecture is in a 1998 article
co-authored by the same R.L.Graham:

Graham RL, Lubachevsky BD, Nurmela KJ, Ostergard PRJ. Dense packings of congruent circles in a circle. Discrete Math 1998;181:139–154.