  Inspired by recent developments on scattering equations, we present a
constructive procedure for computing symmetric, amplitude-encoded, BCJ
numerators for n-point gauge-theory amplitudes, thus satisfying the three
virtues identified by Broedel and Carrasco. We also develop a constructive
procedure for computing symmetric, amplitude-encoded dual-trace functions (tau)
for n-point amplitudes. These can be used to obtain symmetric kinematic
numerators that automatically satisfy color-kinematic duality. The S_n symmetry
of n-point gravity amplitudes formed from these symmetric dual-trace functions
is completely manifest. Explicit expressions for four- and five-point
amplitudes are presented.
