Parity Explained Part 1: Examples and Definitions

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• published: 17 May 2012
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Anyone who's made that fateful attempt to solve a 4x4 after conquering a 3x3 comes right up against this strange and scary concept! But what exactly is it? Why is it there? Here is my attempt at broadly characterizing it. This is NOT an exhaustive explanation of all the characteristics of parity from a mathematical perspective, but more an illustration into what a specific type of parity is in the confines of puzzle solving. Parity really just refers to whether a permutation allows an even or an odd number of swaps, and the parity properties of a puzzle describe what is allowed to get the puzzle to a solved state. I define a PARITY PROBLEM as a situation that seems to break the laws of parity in that puzzle, giving the puzzle an impossible configuration that CAN'T be solved as an odd puzzle, given what you've reduced the puzzle to. So when reducing an even numbered puzzle to an odd numbered puzzle, you may have reduced it in a configuration that it could never have been scrambled into if you started out as an odd puzzle. This is not recognized until the last layer when an impossible configuration is seen. This necessitates breaking your reduction by dissecting into the reduced edges, taking them out of the reduced state, and using an algorithm to redo it. This I distinguish from the Fallacy of False Equivocation, which is not the same kind of PARITY PROBLEM, as no reduction from a puzzle with different parity properties occurred, no illegal moves were made when scrambling, and no breaking of parity occurred, such as disassembly. It only seems like it did because one piece was rotated the wrong way, or was mistaken for another identical one. The solution to this does NOT necessitate dissecting into a reduced edge or disassembling the puzzle, as many viewers have said they had to do in those situations. This CAN be solved as an odd layered puzzle by reorienting a single piece or exchanging two identical pieces with the usual odd puzzle algorithms that swap an expected even number of pieces. Perhaps you can call this a kind of parity, I prefer to distinguish it as a mistaken piece. Coming up, in the next two parts I do example tutorials that demonstrate both of these concepts. Part 2 will be a demonstration of the Fallacy of False Equivocation in the context of a Shepherd's cube tutorial, Part 3 will show true parity from the reduction of the centers of the professor pyraminx to the master or JIng's pyraminx. Stay tuned, and all opinions and perspectives are welcomed!