The original puzzle Two diagonals on the sides of a cube meet in a corner. What is the size of the angle between them ?
The Solution This is a real beauty ! The answer to the original question is not too difficult to find, but it leads to all sorts of new questions concerning volume.
The Analysis Each of the 2 given diagonals, on adjacent faces of a cube, has 2 ends  one end at which they meet, and another open, "loose" end. If a line is drawn between these 2 "loose" ends, it will be another (third) diagonal on a face adjacent to the other 2 faces. These 3 diagonals form an equilateral triangle, giving us the answer to our puzzle.
Mathematics This equilateral triangle of diagonals, acts as the base of a tetrahedron, whose sides are 3 rightangle isosceles triangles, each occupying half of a cube face, 1.5 cube faces in all. There are 4 such tetrahedrons occupying the total surface of the 6 faces of the cube. There also seems to be a fifth tetrahedron, this one perfectly regular, on the inside, enclosed in the 4 others. How is the volume of the cube distributed among these 5 tetrahedrons ?
Pedagogy It might be fun for the children to cut up a softplastic (or styrofoam) cube, along the six diagonals of the cube (one on each face), to remove 4 pyramids from the corners (all built on an equilateral base with 3 isosceles rightangle sides), and discover another pyramid "hidden" inside (perfectly equilateral, base and sides), larger than the other 4 exterior pyramids.
