  A pulse traveling on a uniform nondissipative chain of $N$ masses connected
by springs is soon destructured by dispersion. Here it is shown that a proper
modulation of the masses and the elastic constants makes it possible to obtain
a periodic dynamics and a perfect transmission of any kind of pulse between the
chain ends, since the initial configuration evolves to its mirror image in the
half period. This makes the chain to behave as a Newton's cradle. By a known
algorithm based on orthogonal polynomials one can numerically solve the general
inverse problem leading from the spectrum to the dynamical matrix and then to
the corresponding mass-spring sequence, so yielding all possible ``perfect
cradles''. As quantum linear systems obey the same dynamics of their classical
counterparts, these results also apply to the quantum case: for instance, a
wavefunction localized at one end would evolve to its mirror image at the
opposite chain end.
