[ Concerning the sphere ] $[$ Regular tetrahedron $]$

Four spheres of radius 1 touch each other pairwise. Find the radius of the sphere that touches all four spheres.
Let a sphere of radius $r$ with center $O$ touch internally the given spheres with centers $O_1, O_2, O_3$, and $O_4$. Then $O$ is the center of the regular tetrahedron $O_1 O_2 O_3 O_4$, with edge length 2. The radius $R$ of the sphere circumscribed around the regular tetrahedron $O_1 O_2 O_3 O_4$ is $\frac{3}{4}$ of its height, i.e.,

$$
R=\frac{3}{4} \cdot 2 \sqrt{\frac{2}{3}}=\sqrt{\frac{3}{2}}
$$

Therefore,

$$
r=R+1=\sqrt{\frac{3}{2}}+1
$$

If the sphere of radius $r$ with center $O$ touches the given spheres externally, then similarly we find that

$$
r=R-1=\sqrt{\frac{3}{2}}-1
$$