Let me back up a little bit. My sister and I like the Rubik's Cube, but we don't want to learn algorithms. We don't want to solve it quickly. We don't want to solve it efficiently. We don't want to memorize the patterns of colors and commit them to memory and play with blindfolds. We don't want to impress people. We sure as hell don't want to compete. But we like the Rubik's Cube and we want to be able to solve it.
When we first realized this, we started playing with a standard Rubik's Cube, which has dimensions 3x3x3. Then we learned how to solve a 2x2x2, which is trickier because every move cycles through more of the cube. But that was fun. Then we played with the 5x5x5, which is a lot like the 3x3x3, but it's easier to see what's going on, and of course, it takes longer. We concluded that the 5x5x5 was the best learning tool out of the bunch, with the exception that it had a parity error. In other words, you can be minding your own business like any good citizen, and you may, in a very rude awakening, stumble upon something that isn't intuitively solvable. At this point, you can just mix up the cube and hope it doesn't happen again, cheat with algorithms, or figure it out if you are really smart. Or you can be like the Kratzke sisters and decide that the 5x5x5 Rubik's Cube is no longer a good teaching tool, and furthermore, it is no longer even a good puzzle. In fact, it's no good for anything at all. You could demote the 5x5x5 Rubik's Cube to something that is unworthy of ownership entirely. And if you were to happen to be in that boat, then you would be rather glad to have come across this post. Because, my darlings, if you have made it all the way down here to the end of this section, then it is you to whom I am writing. My sister solved the 5x5x5 parity error.
In order to continue, I'm afraid I need to back up again. First I'll briefly explain how to solve a 3x3x3. Consider the center squares of each of the 6 faces to be set in stone. It will be in your best interest to solve the 12 edge pieces first and then the 8 corner pieces, each of which has 3 sides. Choose a face, defined by its center square, and solve the 4 edge pieces of that face. You may say, "I have not played with a Rubik's Cube before! How will I do that?" You'll see. You can.
Next, solve 3 of the edges that belong to the 4 faces that touch the edge-solved face. Do them one at a time. It will take a little practice, but you can get there. Trust me. Remember to ignore all your corner pieces.
At this point, you have 5 unsolved edges. Your next job is to orient the remaining edge pieces correctly for your final face, and then solve them. It's easier said than done, but not impossible. If you're stuck here, come for a visit and I can help. Once you get past this step, you have to swap corner pieces around. This part is a little tricky, but it makes sense once you understand it. This may require another visit.
If you've made it this far, you've unlocked the achievement where we return to the 5x5x5. This is the same puzzle as the 3x3x3 was, but now, each center piece from the 3x3x3 becomes a 3x3 face, and each edge piece from the 3x3x3 becomes a stack of 3 neighboring edge pieces. The corner pieces are the same as before. To solve the 5x5x5, it is easiest to first solve all the centers, and then all the edges, and then treat the cube as if it were a 3x3x3.
So first, solve any 9-piece center square. From this point, the easiest approach is to solve the center square on the opposite side of your solved center, because then you can solve the remaining 4 center squares without messing up those first 2 solved squares again.
The next step is to solve the 3-piece edges, and there are 12 of them. First, solve all 12 2-piece edges, which may require your third visit. Then complete all 12 3-piece edges. Right? No! Only sometimes right!! This is where a parity error is possible. One of the many unfortunate truths of this world is that you may not be able to convert your 12 2-edge pieces into 12 3-edge pieces without trickery. You may only be able to convert 10 of them. Well, worry no more, because my sister fixed the problem in a way that does not require algorithms, and now we're back to where we started at the beginning of the post. Now is the time for the punchline.
After you have solved 6 9-piece center squares, 10 3-piece edge stacks, and 2 2-piece edge stacks, you're staring straight into the face of a parity error. Position these annoying edges on opposite sides of one of your centers, because it's time to re-solve that center in such a way that solving the edges will be possible with no smoke and no mirrors. Once your problem edges are on opposite sides of a single face, the unsightly pieces will be contained in one 5-piece column. There will be 3 columns in your 9-piece center: the problem column, the middle column, and the other column. Switch the problem column with the other column. Now re-solve everything you just messed up without touching the 9-piece center you just fixed, starting with the 9-piece centers and then the 2-piece edges, and finally the 3-piece edges.
And everything should be just fine.
I'm sweet. I'm sweet.
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