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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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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… – Willie Wong Jul 22 '10 at 0:08
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! – Gmackematix Jul 22 '10 at 0:20
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. – Willie Wong Jul 22 '10 at 10:11
I'd suggest retitling the question since "circular argument" usually doesn't refer to an argument about circles! – Gabe Cunningham Jul 22 '10 at 14:12
up vote 2 down vote accepted

No, according to the information at

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Smallest known doesn't mean impossible. – Willie Wong Jul 22 '10 at 0:13
@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.... – Gerry Myerson Jul 22 '10 at 0:34
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...". – Willie Wong Jul 22 '10 at 10:09
@Willie: point taken. – Gerry Myerson Jul 22 '10 at 11:57

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