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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 τ 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. © 2014 The Author(s).