By using a regular hexagonal arrangement it is simple to fit 19 identical circles into a larger circle of five times the radius with no circles overlapping. This leaves an area equal to six smaller circles uncovered. Is it possible to rearrange the 19 circles to accommodate a twentieth circle of the same size into the larger one (also with no overlapping of course)?
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$\begingroup$ I am not completely convinced that this question is of the right level of discussion for this website; but I am not convinced either that it is not. Can you provide more motivation for the problem? (Is this known to be open, historically interesting, or is this just a puzzle? In the last case this question should be asked at Art of Problem Solving, not here.) I've started a meta thread here tea.mathoverflow.net/discussion/533/… $\endgroup$– Willie WongCommented Jul 22, 2010 at 0:08
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$\begingroup$ I apologise if the question is of the wrong type for this site, but having failed to find any way of answering it myself I Googled a page which led me here. It has no use unless I was trying to market an executive toy that asked people to try and slot in a twentieth circle! $\endgroup$– GmackematixCommented Jul 22, 2010 at 0:20
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$\begingroup$ Like I said, I don't know whether it is appropriate, because I am not sure (until I saw Gerry's answer) whether this problem has research interest at all! It'd be better for you (in the future) to pre-emptively squash those qualms by including more background or describing what you've found or why you've come across the question. $\endgroup$– Willie WongCommented Jul 22, 2010 at 10:11
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4$\begingroup$ I'd suggest retitling the question since "circular argument" usually doesn't refer to an argument about circles! $\endgroup$– Gabe CunninghamCommented Jul 22, 2010 at 14:12
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1 Answer
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No, according to the information at https://erich-friedman.github.io/packing/cirincir/
answered Jul 22, 2010 at 0:09
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$\begingroup$ Smallest known doesn't mean impossible. $\endgroup$ Commented Jul 22, 2010 at 0:13
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$\begingroup$ @Willie, of course, you're right. But if the record for 20 circles has stood at 5.122 since 1971, and we'd need to get that down to 4.863 for an affirmative answer.... $\endgroup$ Commented Jul 22, 2010 at 0:34
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3$\begingroup$ I'm just being pedantic. I would've phrased the answer more along the lines of "According to the information at blah, it is highly unlikely (the current bound has stood for 4 decades!), but no actual proof exists" or something like that. The OP, after all, did ask "Is it possible..." rather than "Is there a known way...". $\endgroup$ Commented Jul 22, 2010 at 10:09
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$\begingroup$ @GerryMyerson for the present question we'd need to get it down to 5.000, not 4.863. That's still a long way. $\endgroup$ Commented Dec 7, 2020 at 15:47