  In this paper we present the results of our calculations of the Einsteinian
strengths S_E(d) and numbers dynamical degrees of freedom N_{DF}(d) for
alternative gravity theories in d >= 4 dimensions. In the first part we
consider the numbers S_E(d) and N_{DF}(d) for metric-compatible and quadratic
in curvature (or quadratic in curvature and in torsion) gravity. We show that
in the entire set of the metric-compatible quadratic gravity in d >= 4
dimensions the 2-nd order Einstein-Gauss-Bonnet theory has the smallest numbers
S_E(d) and N_{DF}(d), i.e., this quadratic theory of gravity has the strongest
field equations. From the physical point of view this theory is the best one
quadratic and metric-compatible theory of gravity in d >= 4 dimensions.
