Feynman's puzzle and the twins: applying the equivalence principle and the red shift

Two puzzles, one not well known and one notorious, are answered exactly and in an elementary fashion with the help of the gravitational red shift and the equivalence principle. The first is Feynman’s question: What motion maximizes the time recorded by a clock moving vertically, if it leaves the earth and returns at a fixed terrestrial time? The second is the supposed inconsistency in the ages of two twins reunited after one has made a round trip at relativistic speed to a distant star