Partition Analysis is a term coined by MacMahon for its algorithmic approach to the construction of explicit solutions of diophantine systems. This algorithm has recently been implemented in MAPLE and MATHEMATICA. After a brief introduction to the algorithm we show its power and limitations in its application to a remarkable diophantine system. Our task is to enumerate the number of ways of assigning weights to the vertices of the Hypercube so that all the maximal faces carry the same weight. Some solutions to this problem are given and its intimate connection to Representation Theory will be also be discussed.