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Saturday, November 19, 2011

The Conquest Of The Rubik's Cube

My daughter goes to school in a school bus. One day, she came back from school and told me that another student on her bus works on a Rubik's cube on the bus. She watches him solve the cube quite often, and her interest was piqued enough that she wanted to play around with a Rubik's cube herself.

So, I took her to a store and got her a new Rubik's cube. I also sat with her and taught her the general principles behind solving one side of the cube. I taught her how to move individual cubes out of the way, bring down a layer, slot the moved cube back into the layer, and then move the layer back to its original position. I also taught her how to get cubes out of the bottom layer or cubes which were stuck on the sides of the top layer (assuming she was trying to solve the top side). My daughter caught on quite quickly, and could solve one side of the cube without any problems. And then started asking me how to solve the whole cube.

The Rubik's cube first came out when I was in school, and all I ever did was learn how to solve one side of it. When I went to college, one of my friends knew how to solve all 6 sides of the cube. I sat with him a few times, and documented the procedure for solving all 6 sides in a notebook (remember, this was before the days of the internet, so I couldn't just Google for the solution technique at that time).

The procedure seemed complicated enough that I never really solved the cube properly using the procedure. I certainly could not memorize the procedure and all the variations and exceptions. After a while, I did not even have access to a cube anymore, and my experimentation with it was long-forgotten. And then I lost the notebook itself, and the cube was a challenge that went unconquered.

Now that I had access to a Rubik's cube once more, my curiosity was re-ignited. The cube came packaged with a solution technique that used simple pictures and a very understandable notation for describing which layers of the cube to turn in what directions. I was actually able to follow the directions and actually solve the cube after the first few attempts (the first couple of times I tried it, I misunderstood one of the steps, and kept messing up the cube).

Rubik's CubeAfter those first few attempts, I took the time to memorize the directions using some simple mnemonics. And now, I can solve a fully scrambled Rubik's cube without referring to any directions or instructions at all. This set of instructions is not optimized, and so you have to use the same set of instructions multiple times to accomplish different steps in the solution process. I am sure there are better instructions out there that are optimized for the specific arrangement of cubes you find yourself facing. But, those instructions are probably more complicated and require more memorization.

In any case, from a fully scrambled state, I can get to a fully solved state in between 5 minutes and 10 minutes. I started out taking just over 10 minutes, but as I have become more familiar with the instructions and how to identify patterns in the arrangement of the cubes, I have improved somewhat. Obviously, I am not setting any records here. It seems more like magic than talent when I hear that the record for solving a cube fully is 29 seconds! 29 seconds!! I can't even solve one side of a cube in 29 seconds if my life depended on it!!!

I can solve one side in about 30 seconds to a minute if I didn't care about the sides. If I wanted to get the first layer of the sides solved along with the top side, then it takes me 90 seconds to 3 minutes. But, I don't really care. The Rubik's cube is one of those timeless puzzles that has fascinated me since my school days. To be able to solve one in less than 10 minutes seems like such an accomplishment that I am tickled pink by it!

When I initially played with a Rubik's cube, I never believed it possible that I would ever be capable of solving it fully without referring to instructions step by step. To be able to now solve it without any help from anybody or anything feels as if I have accomplished something that I could only dream of before. It is like getting a black belt in cube-solving! I can even solve the cube while carrying on a conversation with somebody else.

My conquest of the cube was inspired by my daughter's curiosity. She is still at a point in my life when she considers me smart enough to do anything I want in life. So, I could hardly let her down when she started asking me to teach her how to solve the cube fully. After becoming familiar with the instructions, I started teaching her how to do it.

The first few times, I helped her by reciting the instructions I had memorized at each step as she turned the cube and solved it. After hearing me repeat the instructions several times, they started becoming embedded in her brain, and pretty soon, she could repeat the instructions herself without any help from me. Now, she can also solve the cube in about 5 to 10 minutes without any external help.

I am now trying to get my other daughter to take up the cube as a challenge. She can solve one side right now in about 5 minutes, and solving the top with the first layers of the sides takes her longer. But, I am confident that she will eventually get interested enough to get beyond that to the full solution. I am working at keeping her interest level up so that she does not give it up in the initial stages. That is when things always seem way too hard, and the end result does not seem worth the trouble. We will see how it goes.

Most of the steps in solving the cube are related to isolating different parts of the cube and doing a set of twists and turns that rearranges cubes in a part of the cube without affecting other parts of the cube. I refer to this as isolation. You have to understand isolation and how to accomplish it to be able to go beyond memorizing instructions to solve the cube. If you want to come up with your own solution techniques, you have to figure out how to isolate parts of a cube so that you can rearrange cubes to the required positions while not destroying the arrangement of cubes in other parts of the cube.

To illustrate isolation, I gave my kids a puzzle that you might also find interesting. Take a 4x4 grid of 1's and 0's as below. It doesn't matter where the 1's and 0's are. The important thing is to start with this grid, and do a set of "manipulations" that result in a grid that has just one of those numbers changed from a 0 to a 1 or vice versa. So, consider the two grids below, for instance:

1 0 1 1
0 0 1 0
0 0 1 1
1 1 0 1

1 0 1 1
0 0 1 0
0 0 0 1
1 1 0 1

The second one is just the first one with the 1 in position (3,3) changed to a 0 (I have bolded the elements to show you what has changed). All the other elements are the same. Now, the rules of manipulations are very simple. You can flip any number in the grid from a 0 to 1 or vice versa. When you flip any element in the grid, all the elements in the grid in that row and column also flip (0's change to 1's and 1's change to 0's).

That is all there is to it. Using this rule, come up with a set of manipulations (flips) that change just the element in (3,3) from a 1 to a 0. That is what isolation is all about (though I would daresay, the isolations in a Rubik's cube are a lot more complicated and require a lot more spatial orientation skills). I will provide the solution to this simple puzzle next week. Good luck!

1 comment:

BxCapricorn said...

I did this on my blog recently to show how effective mnemonics can be. I realize your post is a few years old, but maybe you'd like to see my method.

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