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A pulse traveling on a uniform nondissipative chain of $N$ masses connectedby springs is soon destructured by dispersion. Here it is shown that a propermodulation of the masses and the elastic constants makes it possible to obtaina periodic dynamics and a perfect transmission of any kind of pulse between thechain ends, since the initial configuration evolves to its mirror image in thehalf period. This makes the chain to behave as a Newton's cradle. By a knownalgorithm based on orthogonal polynomials one can numerically solve the generalinverse problem leading from the spectrum to the dynamical matrix and then tothe corresponding mass-spring sequence, so yielding all possible ``perfectcradles''. As quantum linear systems obey the same dynamics of their classicalcounterparts, these results also apply to the quantum case: for instance, awavefunction localized at one end would evolve to its mirror image at theopposite chain end.

Dispersionless pulse transport in mass-spring chains: All possible perfect Newton's cradles