  A $d$-dimensional simplex $S$ is called a $k$-reptile (or a $k$-reptile
simplex) if it can be tiled by $k$ simplices with disjoint interiors that are
all mutually congruent and similar to $S$. For $d=2$, triangular $k$-reptiles
exist for all $k$ of the form $a^2, 3a^2$ or $a^2 + b^2$ and they have been
completely characterized by Snover, Waiveris, and Williams. On the other hand,
the only $k$-reptile simplices that are known for $d \ge 3$, have $k = m^d$,
where $m$ is a positive integer. We substantially simplify the proof by
Matou\v{s}ek and the second author that for $d=3$, $k$-reptile tetrahedra can
exist only for $k=m^3$. We then prove a weaker analogue of this result for
$d=4$ by showing that four-dimensional $k$-reptile simplices can exist only for
$k=m^2$.
