a big circle has radius 5 cm is cut down into smaller circles of radius 1 cm .How many maximum number of smaller circle possible? How it is calculated?
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What did you try to solve this problem? – TZakrevskiy Jan 07 '14 at 10:28
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The answer should be $18$, but I wouldn't know how to prove formally it is the best possible packing. – Daniel Robert-Nicoud Jan 07 '14 at 10:50
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@DanielRobert-Nicoud the short trick is 0.83*R2/r2-1.9 but i want to know the concept behind this formula... – anil Jan 07 '14 at 10:56
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@Daniel Robert-Nicoud and anil: There is a better solution – Henry Jan 07 '14 at 10:59
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@anil. Where did you find this short trick ? – Claude Leibovici Jan 07 '14 at 11:20
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@Henry OOps, I miscounted. I meant exactly your solution. – Daniel Robert-Nicoud Jan 07 '14 at 11:29
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@Claude Leibovici somewhere on the internet..I do not remember exactly.... – anil Jan 07 '14 at 11:52
1 Answers
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Erich Friedman's Packing Center suggests that F. Fodor proved in 1999 that $19$ unit circles fit in a circle of radius $1 + \sqrt2 + \sqrt6 =4.863\ldots$, while Goldberg found in 1971 that $20$ unit circles could fit in a circle of radius $5.122\ldots$.
So the answer seems to be $19$.
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is there any generalized concept...suppose if i want to fit 100 same sized circle in a big circle..then what will be the radius of big circle..????? – anil Jan 07 '14 at 10:59
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There will be approximations, but there does not seem to be a general rule. It may be worth reading: Graham RL, Lubachevsky BD, Nurmela KJ,Ostergard PRJ. Dense packings of congruent circles in a circle. Discrete Math 1998;181:139–154. http://www.math.ucsd.edu/~ronspubs/98_01_circles.pdf – Henry Jan 07 '14 at 11:05