  A d-dimensional simplex S is called a k-reptile if it can be tiled without
overlaps by simplices S_1,S_2,...,S_k that are all congruent and similar to S.
For d=2, k-reptile simplices (triangles) exist for many values of k and they
have been completely characterized by Snover, Waiveris, and Williams. On the
other hand, for d > 2, only one construction of k-reptile simplices is known,
the Hill simplices, and it provides only k of the form m^d, m=2,3,.... We prove
that for d=3, k-reptile simplices (tetrahedra) exist only for k=m^3. This
partially confirms a conjecture of Hertel, asserting that the only k-reptile
tetrahedra are the Hill tetrahedra. Our research has been motivated by the
problem of probabilistic packet marking in theoretical computer science,
introduced by Adler in 2002.
