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:

enter image description here

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.