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Table in my room is round in shape and its radius is 15 times the radius of our plates, which are also round in shape. Find the number of plates that can be placed on the table so that they neither overlap each other nor the edge of the table ?

MY Solution :- let the radius of plates be $r$ Then radius of table is $15 r$

Number of plates = $225πr^2/πr^2$ = $225$

I have doubt that is my solution is correct or not ?

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  • $\begingroup$ I’m voting to close this question because this is an open problem: en.wikipedia.org/wiki/Circle_packing_in_a_circle $\endgroup$
    – Mike Earnest
    Commented Feb 26, 2022 at 18:12
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    $\begingroup$ @MikeEarnest Why is being an open problem a reason to close? It's a reason to post an answer saying that the problem is open. $\endgroup$
    – Misha Lavrov
    Commented Feb 26, 2022 at 18:13
  • $\begingroup$ Please don't this sir $\endgroup$
    – Tips
    Commented Feb 26, 2022 at 18:14
  • $\begingroup$ @MishaLavrov You have changed my mind, I retracted the close vote. $\endgroup$
    – Mike Earnest
    Commented Feb 26, 2022 at 18:20
  • $\begingroup$ Still, when people present a question as "Find the solution to _____", it seems to me the poster is tacitly implying the problem has a solution. If you do not know a problem has a solution, part of establishing context should be something like "I came across this problem in ______, does it have a solution?" Please keep that in mind for the future, @tips $\endgroup$
    – Mike Earnest
    Commented Feb 26, 2022 at 18:22

2 Answers 2

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The area of $225$ plates equals the area of the table exactly. So we could only place the $225$ plates on the table if there were no gaps between them. This would be possible with square plates on a square table, but it is impossible to arrange round plates in a manner that does not leave holes.

The exact answer to this problem is open, and it is open even for much smaller problems; http://hydra.nat.uni-magdeburg.de/packing/cci/ summarizes the state of the art. See also Wikipedia's article on circle packing in a circle.

In particular, as shown below:

  • We know a way to arrange $187$ plates on a table with radius slightly above $14.989$ times the radius of a plate. So we could also arrange those $187$ plates on a table with radius $15$ times the radius of a plate.
  • The best arrangement of $188$ plates known requires a table with radius more than $15.028$ times the radius of a plate. So it cannot be implemented here; to put $188$ (or more) plates on your table would require a better solution than what is currently the state of the art.

enter image description here

(The images are also taken from http://hydra.nat.uni-magdeburg.de/packing/cci/)

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  • $\begingroup$ Very nice, +1. I just wanted to remark that (though it seems likely) it doesn't strictly follow from these values being the state of the art, that the solution to this exact problem is unknown $\endgroup$
    – doetoe
    Commented Feb 27, 2022 at 8:33
  • $\begingroup$ Right, we could imagine that someone proved a lower bound of $15.001$ on the radius of the table for $188$ circles. But I'm pretty sure we know way less than that about any of the problems. $\endgroup$
    – Misha Lavrov
    Commented Feb 27, 2022 at 15:22
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You have correctly obtained an upper bound, but the best known lower bound seems to be $187$: http://hydra.nat.uni-magdeburg.de/packing/cci/d16.html

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