This study explores so-called Darwinian Daisyworlds mathematically rigorously in detail. The original Daisyworld was introduced by Watson & Lovelock (1983) to demonstrate how two species of daisies regulate the global temperature of their planet through competition among these species against the rising solar luminosity, i.e. the Gaia hypothesis. Its variants are Darwinian Daisyworlds in which daisies can adapt themselves to the local temperature. Robertson & Robinson (1998) insist their Darwinian daisies lose the ability for temperature regulation on the basis of their spreadsheet simulations. Lenton & Lovelock (2000) point out that the constraints on adaptation recovers Darwinian daisies' ability of temperature regulation on the basis of their Euler-code simulations. The present study shows there exist the exact and closed-form solutions to these two Daisyworlds. The results contradict the former studies: Robertson and Robinson's daisies do regulate the global temperature even longer than non-adaptive daisies; Lenton and Lovelock's daisies are less adaptive than Robertson and Robinson's daisies because of the constraints on adaptation; the introduction of weak adaptability drives species into a dead end of evolution. Thus, the present results confirm that the Gaia hypothesis and Darwinian evolution can coexist.