Box stacking problem

Question

I found this famous dp problem in many places, but I can not figure out how to solve.

You are given a set of n types of rectangular 3-D boxes, where

Answer 1

The stack can be regarded as a sequence of x,y,z triplets (x,y being the 2D plane, and z the height), where x(i) > x(i+1) and y(i) > y(i+1). The goal is to maximize the sum of z picking the triplets from the set of available triplets - each triplet being one type of box in a particular orientation. It is pretty easy to see that enforcing the constraint x > y doesn't reduce the solution space. So each box generates 3 triplets, having each of w,h,d as the z coordinate.

If you regard the triplets as a directed acyclic graph where edges of length z exist between two triplets when the x,y constraints are satisfied between them, then the problem is of finding the longest path through that graph.