Puzzle 1
    

Figure 1                                                                          Figure 2

The Puzzle:

    What's the largest cube that can be inscribed inside a sphere with a radius of 1 unit?  Give the answer in terms of how long the edge of the cube must be in radius units.

    Contrarily, assume the edges of the cube to be 1 unit long, then what radius (or diameter) will the sphere have in edge units?

Hint:  The diagonal of a square is the length of an edge multiplied times the square root of 2.  Thus, if an edge is  1 unit long, the diagonal of that sqaure will be _/^~2, as in Figure 1.

Hint2:  The answer to the puzzle is NOT _/^~2 !
 

    What's the largest sphere that can be inscribed inside an Equilateral Tetrahedron with edge lengths of 1 unit?  Give the answer in terms of how long the sphere radius must be in edge lengths.

Hint 1:  The sphere will be tangent to all four tetrahedral surfaces at the altitude points of the respective triangular surfaces, as indicated by the blue dots in Figure 2 above.

Hint 2:  The slope angle between two faces is NOT the same as the angle between a face and an edge.