+2489 Well, I can bloodywell state with near certainty that VooFix doesn’t understand it at all!
One reason for this certainty is this is NOT the solution to the question he posted. The posted question has g(6) = -133, the provided solution is correct for g(6) = -113. So strike one for even paying attention to the question, let alone understanding the solution.
This solution uses a basic generating function (and it’s very cool). I offer brief comments on generating functions here and here. Generating functions can be very complex, but even understanding the basic, rudimentary generating functions require at least a moderate prerequisite understanding of Church-Rosser properties for the underlying equivalence relations, and a working knowledge for converting augmented matrixes to reduced row echelon form via Gauss or Gauss–Jordan Elimination.
Though it’s unlikely (without further, detailed instruction) that any of the AoPS students presented with this solution will be able to casually use this generating function to solve this problem, there are many who could (without understanding the underlying fundamentals) learn to solve this and some similar problems using generating functions with rote instructions.
Most students (the very vast majority) learn the rote procedures first, and then the understanding of the why (maybe) comes during the next level. This is true for both adult students and children. Consider elementary students learning how to multiply: In the beginning, most do not know they are adding a number to itself a certain number of times. For some students this understanding comes when they learn to divide. Of course, some will never understand this, but they can still multiply numbers.
Instruction beyond the grasp [of understanding] does seem to be an ideal method for teaching; and reaching beyond ones grasp also seems to be an ideal method for learning. This is why AoPS used a generating function for the solution. AoPS will introduce a concept with a basic outline of the fundamentals and then present a solution that is at least two levels higher than the students’ current skills. From this point, they teach from the top-down and from the bottom-up, and the skills and understanding meet somewhere in the middle. The brighter students will see where they are headed with this, even if they are uncertain of the path. The lesser students may not see where they are going, but they do have bread crumbs to find their way –unless they wait too long and the ants carry the crumbs off. 
This is a very cool teaching method. My great uncle Cosmo often used the top-down-bottom-up scheme to teach me science and the scientific method (and other subjects). My education started at age five. My aunt or uncle would read me a children’s story –usually one of the classics; Cosmo would then read me short passages from a science text. The first of many texts was the Origin of Species by Charles Darwin, which he supplemented with excerpts from the two volumes of The Voyage of the Beagle. These texts weren’t children’s editions –they were originals. After reading a few passages to me, he would explain what it meant. I’m sure I still didn’t understand it. But Cosmo wasn’t worried about it; he knew one day I would understand this theory and that my brain would grow and develop around this foundation of knowledge.
Lancelot Link also used the top-down-bottom-up method to tutor and teach me more than four years worth of collegiate mathematics in less than a year and a half. From childhood, I’ve always known I was an advanced Chimp, but I became a Genetically Enhanced Chimp before I graduated from Lancelot Links School of Mathematics and Trolling. I don’t think I would have passed the classes without the enhancement. ...Or maybe it was passing the classes that gave me the enhancement. 
GA
