# Packing equal-size disks in a unit disk

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?

• A good resource for this kind of question is Erich's Packing Center, erich-friedman.github.io/packing/index.html Apr 2, 2021 at 11:55
• 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. Apr 2, 2021 at 13:42
• 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.
– Hao
Apr 2, 2021 at 19:44
• @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. Apr 3, 2021 at 20:18

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: