The Razumov-Stroganov conjectures arose in statistical physics as a surprising link between the XXZ spin chain at a special value of the parameters and the number of states in a completely different combinatorial problem called the Fully Packed Loop model. This simple-to-state-but-hard-to-prove result has garnered a lot of attention among physicists, combinatorialists and everyone in between since it was first stated in 2001. We formulate the problem from a purely combinatorial point of view and find a new structure which we hope can be exploited towards the correct solution from Erd\H{o}s' book. No previous knowledge will be assumed. (Joint work with Doron Zeilberger)