It is relatively easy to see that the largest regular tetrahedron which will fit inside a cube whose edges have length 1 is the tetrahedron shown in the figure below. The length of an edge of this tetrahedron is Sqrt[2].

This month there are two questions:

- What is the length of an edge of the largest regular tetrahedron which will fit inside a regular octahedron whose edges have length 1?"
- What is the length of an edge of the largest cube which will fit inside a regular tetrahedron whose edges have length 1?"

(GIF's © 1998, R. Mäder)