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Prove that for all circle of radius $3^n$ we can have $7^n$ circles inside with radius of 1 and neither of them intersect.

For me, it sounds like using mathematic induction, but I have no clear idea or answer.

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    $\begingroup$ From $n=6$ onward, even a wastefully loose packing of small circles into $2\times 2$ boxes would suffice ... ($9^n\pi -O(3^n)\gg 4\cdot 7^n$) $\endgroup$
    – Hagen von Eitzen
    Commented Jun 27, 2017 at 14:05

2 Answers 2

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Yes, induction should work. The picture below should give you a hint why:

enter image description here

(the radius of the green circles is three times those of the red ones; the radius of the blue circle is three times those of the green ones)

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First, prove the base case (should not be terribly difficult to show geometrically). Here's the idea for the induction step: consider a circle of radius $3^{n+1}$. How many circles of radius $3^n$ can you fit inside? How many circles of radius $1$ can you fit inside each of the circles of radius $3^n$?

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