# Necklaces made out of circles lying on vertices of regular polygons

A size $n$ necklace consists of $n$ touching circles arranged in a regular polygon.

What's the largest necklace that fits inside size $n$ necklace?

For example, $10$-necklace placed inside $17$-necklace:

Is it true that $n-7$ is the largest size that fits inside size $n$, for all $n\ge10$?

I know that if $s$ is the radius of a single circle, then the radius of circle that connects the centers of all circles on a $n$-necklace is (radius of circumscribed circle of regular $n$ sided polygon with side length $2s$): $$\frac{s}{\sin{\frac{\pi}{n}}}$$

Which makes the radii of inscribed and described circles of a $n$ sized necklace: $$\frac{s}{\sin{\frac{\pi}{n}}}-s,\frac{s}{\sin{\frac{\pi}{n}}}+s$$

Which means one needs to find the smallest $m$ such that

$$\frac{s}{\sin{\frac{\pi}{n}}}-\frac{s}{\sin{\frac{\pi}{n-m}}}-2s\ge0$$

Which I suspect to be $m=7$ for all $n\ge10$, but I'm not sure how to prove it?

For $n$ large enough, $\sin \frac{\pi}{n}\approx \frac{\pi}{n}$. You can make this precise by using higher order approximations based on the power series of $\sin$. But using this rough estimation, we have $\frac{1}{\frac{\pi}{n}}- \frac{1}{\frac{\pi}{n-m}} -2 > 0$, so $\frac{m}{\pi}>2$, thus $m> 2\pi$. So for large enough $n$, you are surely right. Compute with higher precision after cheking the first couple of initial values, and you are done.