253. Three identical cylindrical surfaces of radius $R$ with mutually perpendicular axes touch each other pairwise.

a) What is the radius of the smallest sphere that touches these cylindrical surfaces?

b) What is the radius of the largest cylinder that touches the three given ones, the axis of which passes inside the triangle with vertices at the points of contact of the three given cylinders?
253. Consider a cube $A B C D A_{1} B_{1} C_{1} D_{1}$ with edge $2 R$. Place the axes of the given cylinders on the lines $A A_{1}, D C, B_{1} C_{1}$.

a) The center of the cube is at a distance of $\dot{R} \sqrt{2}$ from all edges of the cube. Any point in space is at a distance greater than $R \sqrt{2}$ from at least one of the edges $A A_{1}, D C, B_{1} C_{1}$. This follows from the fact that the cylinders with axes $A A_{1}, D C, B_{1} C_{1}$ and radii $R \sqrt{2}$ have only one common point - the center of the cube. Therefore, the radius of the smallest sphere that touches all three cylinders is $R(\sqrt{2}-1)$.

b) If $K, L$ and $M$ are the midpoints of the edges $A A_{1}, D C$ and $B_{1} C_{1}$, then the line passing through the center of the cube perpendicular to the plane $K L M$ is at a distance of $R \sqrt{2}$ from the lines $A A_{1}, D C$ and $B_{1} C_{1}$; $\triangle K L M$ is equilateral, its center coincides with the center of the cube. From this, it follows that any line intersecting the plane $K L M$ is at a distance from at least one of the vertices of $\triangle K L M$ not greater than the radius of the circumscribed circle around it, which is $R \sqrt{2}$. Thus, the radius of the largest cylinder that touches the three given cylinders and satisfies the conditions of the problem is $R(\sqrt{2}-1)$.