Draw two circles with the same center, one with a radius of $1 \mathrm{~cm}$ and the other with a radius of $3 \mathrm{~cm}$. In the region outside the circle with a radius of $1 \mathrm{~cm}$ and inside the circle with a radius of $3 \mathrm{~cm}$, draw circles that are simultaneously tangent to both circles, as shown in the figure below.

(a) What should be the radius of these circles?

(b) What is the maximum number of these circles, assuming they do not overlap?

![](https://cdn.mathpix.com/cropped/2024_05_01_eead07372ad548e635bag-05.jpg?height=250&width=252&top_left_y=1214&top_left_x=1424)
(a) Since the circles with radii $1 \mathrm{~cm}$ and $3 \mathrm{~cm}$ are concentric, the other circles shown in the figure must have a radius equal to $1 \mathrm{~cm}$.

(b) The centers of the 3 circles with radius $1 \mathrm{~cm}$ shown in the figure form an equilateral triangle with side length $2 \mathrm{~cm}$. Therefore, their internal angles measure $60^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_01_eead07372ad548e635bag-23.jpg?height=320&width=331&top_left_y=377&top_left_x=1339)

Since $\frac{360}{60}=6$, we conclude that up to 6 circles can be arranged under the given conditions.