  In this paper, we study the relation between the Cachazo-He-Yuan (CHY)
formula and the maximal-helicity-violating (MHV) amplitudes of Yang-Mills and
gravity in four dimensions. We prove that only one special rational solution of
the scattering equations found by Weinzierl support the MHV amplitudes. Namely,
localized at this solution, the integrated CHY formula reproduces the
Parke-Taylor formula for Yang-Mills amplitudes as well as the Hodges formula
for gravitational amplitudes. This is achieved by developing techniques, in a
manifestly M\"obius covariant formalism, to explicitly compute relevant reduced
Pfaffians/determinants. We observe and prove two interesting properties (or
identities), which facilitate the computations. We also check that all the
other $(n-3)!-1$ solutions to the scattering equations do not support the MHV
amplitudes, and prove analytically that this is indeed true for the other
special rational solution proposed by Weinzierl, that actually supports the
anti-MHV amplitudes.
