Suggestion for solution
To solve a problem of this type, we need to construct a model for the analysis. The following statement explicitly tells you how to construct such a model.

Use the diagram shown here as a starting point, and construct a tetrahedral arrangement by placing four spheres of radius R at alternate corners of a cube. Once complete, work out the followin:

• What is the length of the face diagonal fd of this cube in terms of R?
  Since the spheres are in contact at the centre of each cube face, fd = 2 R.
• What is the length of the edge for such a cube, in terms of R?
  Cube edge length a = Ö2 R
  
• What is the length of the body diagonal bd of the cube in R?
  bd = Ö6 R
• Is the center of the cube also the center of the tetrahedral hole?
  Yes, but do you know why?
• Let the radius of the tetrahedral hole be r, express bd in terms of R and r
  If you put a small ball there, it will be in contact with all four spheres. Thus, bd = 2 (R + r). r = (2.45 R) / 2 - R
      = 1.225 R - R
      = 0.225 R
  
• What is the radius ratio of tetrahedral holes to the spheres?
  r / R = 0.225

What is the radius of the largest sphere that can be placed in an octahedral hole without pushing the spheres apart?

Solution
The octahedral hole is located at the center of any four spheres that form a square. If we represent the radius of a ball fitting in the octahedral holes by r, and the radius of the sphere as R, then we have the relationship:

The implication:
Pure geometric consideration shows that only small balls fit in the tetrahedral holes of packed spheres. However, if the radii of cations are smaller than 0.225 R, the structure of having ions in the tetrahedral site is unstable. The anions may be pushed apart slightly to reduce the repulsion by fitting a cation in the tetrahedral site.

For ionic crystal structure consideration, the cations are usually smaller than anions. Cations fitting into the tetrahedral sites cannot be smaller than 0.225 R. Usually, most ions are slightly larger than 0.225 R, but smaller than 0.414 R. In such cases, the cation coordination is tetrahedral, and a typical structure is ZnS, although covalent bonding is also involved in ZnS. The animated diagram is a model of ZnS structure.

When the cation radii are greater or equal to 0.414 R, but less than 0.732 R, the cations occupy the octahedral sites. Sodium chloride is one such structure, and it serves as an important structure type.

If the cations are large such that r > 0.732 R, the cation will have a cubic coordination of 8. The strcture is typified by CsCl.

The above discussion is summarized below:

r/R	0.225	between	0.414	between	0.732	<
Coordination & number	"	tetrahedral 4	"	octahedral 8	"	cubic 8
Typical structure	"	ZnS	"	NaCl	"	CsCl

For an interesting, illustrateive and exciting discussion regarding radius ratio and types of inorganic solids, see Structures of Simple Inorganic Solids by Dr. S.J. Heyes. This link is aimed at a higher level than that of a first year chemistry course, but the content is great.

Discussion
What is the radius of the largest sphere that can be placed in an octahedral hole without pushing the spheres apart?
(Answer: 0.731)

Is there a structure in which all the tetrahedral sites are occupied by a different type of atoms or ions?

Solution
The outline of a unit cell for PuO₂ is shown here, and all the tetrahedral sites are occupied by small O^2- ions.

Actually, the crystal structure of UO₂ has the same structure as PuO₂. A common salt CaF₂ also has the same structure, but the fluoride ions are by no means small compared to the calcium ions. However, the Pu⁴⁺ and U⁴⁺ ions are large compared to the oxygen ions.