ISL 1988 Problem 37

i.) Four balls of radius 1 are mutually tangent, three resting on the floor and the fourth resting on the others. A tedrahedron, each of whose edges has length $s,$ is circumscribed around the balls. Find the value of $s.$

ii.) Suppose that $ABCD$ and $EFGH$ are opposite faces of a retangular solid, with $\angle DHC = 45^{\circ}$ and $\angle FHB = 60^{\circ}.$ Find the cosine of $\angle BHD.$