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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?

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    $\begingroup$ A good resource for this kind of question is Erich's Packing Center, erich-friedman.github.io/packing/index.html $\endgroup$ – Gerry Myerson Apr 2 at 11:55
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    $\begingroup$ Changing the question just because it was answered quickly is a bit unfriendly towards the answerer, and confusing for later readers (who will find answers to the two different questions mixed together) — it’s much better to accept the answer, and then ask the follow-up as a new question. Of course, it’s different if the quick answer is trivial, e.g. pointing out a typo, or an obvious degenerate solution the question forgot to rule out. But here, the original question was interesting and non-trivial, and the answer answered it very well. $\endgroup$ – Peter LeFanu Lumsdaine Apr 2 at 13:42
  • $\begingroup$ Peter: Thanks for the comment. I am relatively new to mathoverflow. Which of the following ways are considered acceptable in this case: (a) posting an entire new question including a link to this one, (b) posting the new question as a comment on the original question, or (c) commenting under Noam's answer? I thought the purpose of having websites like mathoverflow is to facilitate the discussions on interesting math problems. But if there are certain community policies users should stick to, I would be happy to learn about them. $\endgroup$ – Hao Apr 2 at 19:44
  • $\begingroup$ @Hao Supporting Peter's proposal which btw is an agreed procedure in these not so seldom cases, so please go for (a) ! All the more your refined follow-up question merits a new question posting, whereas your (b) & (c) generate comments, being more fugacious in character. $\endgroup$ – Hanno Apr 3 at 20:18
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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.

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    $\begingroup$ Thanks for the quick answer! I checked the the next interesting case (36 balls) on hydra.nat.uni-magdeburg.de/packing/cci. And somewhat surprisingly the best known radius beats the 37 ball case. $\endgroup$ – Hao Apr 2 at 1:58

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