49. How many circles of radius 1 can be applied ${ }^{1}$ ) to a given unit circle $\mathcal{S}$ so that no two of these circles intersect? So that no one of these circles contains the center of another circle inside itself?
49. Since a unit circle tangent to $S$ is seen from the center $O$ of circle $S$ at an angle of $60^{\circ}$ (Fig. $132, a$), no more than $6\left(=\frac{360}{60^{\circ}}\right)$ non-overlapping unit circles can be applied to $S$. Six circles can obviously be applied (Fig. 132, b).

Further, if $O_{1}$ and $O_{2}$ are the centers of two unit circles $C_{1}$ and $C_{2}$ applied to $S$ such that $O_{1}$ does not lie inside $C_{2}$ and $O_{2}$ does not lie inside $C_{1}$ (Fig. $133, a$), then $O_{1} O_{2} \geqslant 1, O O_{1}=$ $=O O_{2}=2$ and $\angle O_{1} O O_{2} \geqslant 2 \arcsin \frac{1}{4} \approx 29.0^{\circ}$. It follows that no more than 12 (12 is the integer part of the fraction $\frac{360^{\circ}}{29.0^{\circ}}$) unit circles can be applied to $S$ such that no circle contains the center of another circle; 12 circles can obviously be applied (Fig. 133, b).

![](https://cdn.mathpix.com/cropped/2024_05_21_962b75a91b7185035ab9g-163.jpg?height=409&width=218&top_left_y=795&top_left_x=222)
a)

![](https://cdn.mathpix.com/cropped/2024_05_21_962b75a91b7185035ab9g-163.jpg?height=590&width=620&top_left_y=641&top_left_x=494)

Fig. 132.

![](https://cdn.mathpix.com/cropped/2024_05_21_962b75a91b7185035ab9g-164.jpg?height=1398&width=764&top_left_y=211&top_left_x=220)

Fig. 133.