1. At each vertex of a cube with an edge length of 1, there is the center of a sphere. All spheres are the same, and each touches three adjacent spheres. Find the length of that part of the spatial diagonal of the cube that lies outside the spheres.
Solution. Consider two spheres, the centers of which lie at the vertices of one edge. Since the spheres have the same radius and touch each other, the radius of each sphere is $\frac{1}{2}$. Consider the section $A B C_{1} D_{1}$. This section contains the diagonal $A C_{1}$, and each sphere cuts off a segment of length $\frac{1}{2}$ from this diagonal. The diagonal of the cube has a length of $\sqrt{3}$, so the length of the segment of the diagonal lying outside the spheres is $\sqrt{3}-\frac{1}{2}-\frac{1}{2}=\sqrt{3}-1$.

![](https://cdn.mathpix.com/cropped/2024_05_06_5dd4e49de06ec5f71426g-1.jpg?height=614&width=534&top_left_y=978&top_left_x=824)

Answer. $\sqrt{3}-1$.