(solved by Dmitry Ivanov). Let $A$ and $B$ be two opposite vertices of a cube with side length 1. What is the radius of the sphere centered inside the cube, tangent to the three faces that meet at $A$ and the three edges that meet at $B$?
. Introduce an orthonormal reference frame such that $A=(0,0,0)$ and $B=(1,1,1)$, with the sides of the cube being parallel to the axes. Let $r$ be the radius of the sphere. Its center is the point with coordinates $(r, r, r)$, and the point of tangency with one of the edges at $B$ is $(r, 1,1)$. The distance between these two points is $\sqrt{2}(1-r)$, from which we deduce $r=\sqrt{2}(1-r)$ and $r=2-\sqrt{2}$.