A box can be packed with a Harmonic Brick Iff the box has dimensions for some natural numbers , , (i.e., the box is a multiple of the brick).
See also Box-Packing Theorem, Conway Puzzle, Klarner's Theorem
References
Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., pp. 69-72, 1976.