How to Read Algorithms When Solving a Rubik's Cube

ike most other children who grew up in the '80s, I owned a Rubik'due south Cube that I couldn't solve. Sure, I could complete i face up of the cube. Occasionally, I would luck into completing virtually of a 2nd face. And so, frustrated that whatsoever farther manipulation would ruin all of my hard work, I did what I assume virtually kids my historic period did — I peeled off stickers and swapped them. The resulting cube would impress my younger brother at first, but later noticing the slightly-skewed stickers with wrinkled edges, he as well learned my secret to "solving" the cube.

Past swapping stickers, my brother and I would unwittingly make solving some faces impossible. It took me a while to realize that two same-colored stickers should never share an edge on the cube, or that having more than iv aforementioned-colored stickers on corner pieces was a bad thought. Information similar that eventually helped me swap stickers more efficiently, but information technology was never enough to help me actually solve the cube.

So, I eventually gave up. My father's confidence in my budding genius probably never fully recovered.

Fast-forward xxx years. I am manning the AoPS table at the almanac national MATHCOUNTS championships. At the tabular array side by side to ours are representatives from Rubik's. They've got cubes, and they've got instructions on how to solve them, and I accept plenty of time. This was my chance at redemption!

Terms and Symbols for Set up

The pamphlet of instructions began with some useful terms and symbols. Edge pieces always take 2 different colors. I learned that the hard mode 30 years earlier. Corner pieces have three unlike colors. I remembered that, too. Center pieces have 1 color, and they don't move relative to each other. That fact had escaped me equally a child. The yellow face is e'er opposite the white face, for case.

Adjacent come more terms and symbols. There are 12 possible quarter-plough moves, named by the face y'all are turning (Right, Left, Back, Front, and the more awkwardly named Downwards and Up faces). A standard turn is clockwise, and a counter-clockwise turn is denoted by an "i" for inverted.

The First Phase

OK, got information technology. On to solving the cube. The first few stages are pretty intuitive. With a lilliputian do, I could confidently solve one third of the cube apace and without memorizing whatsoever special sequences. Information technology's but a matter of trading one piece for another until your cube looks something like this:

As a child, I never actually considered that the colors on each face e'er match the color of the middle square. Y'all don't move the centers, you motion the pieces around them. I was mildly embarrassed that I didn't think of this earlier, but also happy to exist learning something.

The 2d Phase

The next pace is to solve the heart third of the cube. This involves memorizing two sets of sequences that permit y'all to swap pieces from the top layer to the middle layer (without peeling off stickers!) while keeping the solved face intact. I learned to orient the cube properly and memorized both sequences (U, R, Ui, Ri, Ui, Fi, U, F and Ui, Li, U, L, U, F, Ui, Fi) until I could solve two thirds of the cube without peeking at the instructions. All the same, I never really understood why these turns worked and was a little skeptical about my newfound skills. Unlike the previous steps, these sequences were starting to experience like magic. Notwithstanding, I pressed on.

The Tertiary Phase

The next stage of the instructions include three 6-motility sequences and an viii-move sequence that sometimes gets repeated. Each of these sequences is paired with a particular cube orientation. At this point I was just trusting the algorithms blindly and trying to build some muscle retentiveness as I adept the diverse sets of twists and turns.

The Last Stage

Completing the final stage involves memorizing a xiii-move sequence of turns and two 12-move sequences. Missing a motility or turning something the incorrect manner nearly e'er forced me to get back to the showtime. Nevertheless, I somewhen managed to solve the cube several times without the instructions. Later two days of playing with the cube and memorizing the sequences, I got my times down to what I felt was a respectable 2 minutes. (The electric current world tape is less than three.5 seconds!)

Unfortunately, my victory over the cube was pretty hollow.

What Had I Learned?

Here's the thing. I achieved the ultimate goal of solving all vi faces of the cube, but I had almost no understanding of what I was doing. The sequences I memorized didn't have any pregnant to me. Y'all would remember that anyone who could solve all six faces could speedily solve exactly two faces. Nope, not me! You could have a solved cube, make four or 5 random turns, hand it back to me, and I'd barely be improve off than if y'all mixed it up completely.

The only affair I really learned by "learning" to solve the Rubik's cube is that I hadn't learned anything at all. That turned out to be a pretty important lesson.

Figuring out i way to solve the Rubik's cube gave me a glimpse into the heed of a child who is taught math as a set of memorized algorithms without e'er learning the "why?" behind them. Also often, students memorize a prepare of steps that allows them to arrive at a correct answer without e'er actually thinking about what they're doing, why it works, or how to adapt and utilise it to other situations.

This was non a new realization for me. Early on in my teaching career, I tried something with my 8th grade classes that would solidify my pedagogy philosophy. I created a page of 6 questions and asked my students to answer every 1 without writing anything except the respond. All of the work had to be done in their heads. Afterwards, I asked that they flip their papers over and write a brief explanation of how they thought about the trouble. So, we discussed their explanations.

1 question asked students to compute seven×106. Nearly everyone got it correct. However, in some classes, every single 8th grader explained how they mentally practical the standard multiplication algorithm, keeping track of digits as they performed the familiar steps. "First, I imagined the vii under the 106. And so, I did 7×6=42. I imagined the 2 at the lesser, and put the 4 over the 0," and so on. Kids mimed with their hands where the invisible digits would go in each step.

When I explained that they could have simply done 7×100=700, and then added on 7×vi=42 to become 742, many were stunned. Whole classes of kids had never even considered using a method other than the i they had been taught in third grade. Many asked, "Are we allowed to do that!?"

I couldn't believe it. These were students who could use the distributive holding with constants and variables, only who never considered breaking upwards a production to make it easy to compute mentally. Their grasp of multi-digit multiplication was as shallow as my grasp of the Rubik'southward cube.

My experiences with the cube and with my students gave me valuable insights into the difference between knowing and agreement. Everyone seems to agree that educators should "teach for understanding," but non everyone seems to concord on what that means.

When math is taught well, students are asked to confront dubiousness and overcome it on their ain. This encourages real agreement — forth with all sorts of other valuable traits similar patience, curiosity, and resilience. But, parents watching a child struggle through tough problems frequently wonder why students tin't just acquire the "like shooting fish in a barrel" way. In other words, why can't their child but memorize a set of steps that will e'er give them the right answer?

Getting the Right Reply is Important, simply non Enough

Knowing how to perform complex computations quickly and accurately is no longer a particularly useful skill. It should be obvious to anyone carrying a jail cell phone that noesis is not nearly as of import as it was fifty years ago. I will never know as much as I can expect up on my phone. I'll never be able to compute as quickly, either. Being able to recall information, utilise formulas, and perform repetitive algorithms speedily and accurately is a domain that is increasingly dominated by machines. What's useful now is the ability to apply what we know. That requires understanding.

While repeating an algorithm can lead to understanding for some students, information technology's rarely the best manner. For near students, a far more efficient path to agreement comes from discovering the "why?" behind an algorithm before they always meet information technology. Students who are encouraged to figure out the "why?" can make connections, create their own algorithms, and utilize what they've learned in a diverseness of situations.

On the other manus, students who learn math the way I learned to solve the cube are left with a series of  meaningless steps that are hard to remember and piece of cake to mess upward. And at least when I mix upwardly steps while solving a Rubik's cube, I can tell I've done something incorrect; the colors don't match! Students don't get such obvious clues when they brand mistakes on math problems.

The consequences of my shallow agreement of the Rubik'due south cube are trivial, but the consequences for students who have only learned to compute the "easy" way can bear on for years. If an easy way doesn't contribute to agreement, math gets a lot harder in the long run.
I take long since forgotten the sequences of moves required to solve a Rubik'due south cube. There was a time when I knew them, but I never understood them, so they were easily erased from memory. Ane solar day, mayhap I'll get around to truly figuring out the cube. When I exercise, I'll have to write a pamphlet to help others empathise it.

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Source: https://artofproblemsolving.com/news/articles/knowing-vs-understanding-rubiks-cube-difference

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