In this paper I present a conjecture for a recursive algorithm that findseach permutation of combining two sets of objects (AKA the Shuffle Product).This algorithm provides an efficient way to navigate this problem, as eachatomic operation yields a permutation of the union. The permutations of theunion of the two sets are represented as binary integers which are thenmanipulated mathematically to find the next permutation. The routes taken tofind each of the permutations then form a series of associations or adjacencieswhich can be represented in a tree graph which appears to possess someproperties of a fractal. This algorithm was discovered while attempting to identify every possibleend-state of a Tic-Tac-Toe (Naughts and Crosses) board. It was found to be aviable and efficient solution to the problem, and now---in its more generalizedstate---it is my belief that it may find applications among a wide range oftheoretical and applied sciences. I hypothesize that, due to the fractal-like nature of the tree it traverses,this algorithm sheds light on a more generic principle of combinatorics and assuch could be further generalized to perhaps be applied to the union of anynumber of sets.
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