Circles of one integer diameter tightly packed around a circle of another integer diameter Is it true that the only integer circle size you can tightly pack around another integer circle size is when the ratio between the sizes of the outer circles and the inner one is $1:1$? Do any other integer ratios exist, or might there be a proof that there are none?

If there are $n$ circles of radius $r$ around a unit one, by trigonometry

$$\frac r{1+r}=\sin\frac{\pi}n$$ and by Niven's theorem, only $r=1$ can do.