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Mathematics/Probability of Rubik's Cube

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Papes
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I encountered my first Rubik's cube about 3 months ago in the dwelling of my best friend. He was messing around with a Revenge (4x4) and couldn't even get one side. I did some research and discovered that there are 43,252,003,274,489,856,000 possibilities for the cube to be arranged (Wikipedia it for more info). Would somebody like to explain to me how out tiny human brains are able to take all 43 quintillion possibilities and reduce it to one solved cube?

Enlightenment welcome.

P.S. Anyone can solve the cube. My best average of 12 is 58 seconds, which is terrible.

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Mainly because we're able to break the problem down into its basic components ('how do I swap 2 edge pieces' / 'how do I rotate a corner' etc) - you only need to know a handful of different algorithms to be able to solve it.

I used to be able to solve a rubik's cube but I never got into speed cubing and my best time was about 2 minutes or something :lol:

Edited by eriatarka
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Would somebody like to explain to me how out tiny human brains are able to take all 43 quintillion possibilities and reduce it to one solved cube?

Enlightenment welcome.

Well, for one thing a brain that tries to solve a Rubik's Cube by trying out every possibility is tiny indeed. Human beings have rational minds. Therefore we can determine how the cube works and find the way to a solution without trying random possibilities mindlessly.

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  • 4 months later...

But even so, I find it amazing that with that many possible positions, we have managed to find the optimal solutions to every position, even if many, many of the patterns are easily solved by using the same algorithm. By the way, my new best average of 12 is 33.4 seconds.

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  • 2 months later...

One reason is symmetry. A lot of the conversions/transforms are symmetric for all six sides, making it essentially one set of possibilities instead of six. Roughly you can reduce the total combinations by 1/6. Then there are smaller symmetries for rows/columns that reduce the total number. This is how a well programmed computer/machine would approach the problem at least initially.

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I encountered my first Rubik's cube about 3 months ago in the dwelling of my best friend. He was messing around with a Revenge (4x4) and couldn't even get one side. I did some research and discovered that there are 43,252,003,274,489,856,000 possibilities for the cube to be arranged (Wikipedia it for more info). Would somebody like to explain to me how out tiny human brains are able to take all 43 quintillion possibilities and reduce it to one solved cube?

Enlightenment welcome.

P.S. Anyone can solve the cube. My best average of 12 is 58 seconds, which is terrible.

Because the solution doesn't require anything more than an understanding of the algorithm behind the solution. The number of possible arrangements is irrelevant to understanding the puzzle itself. The puzzle itself is 9 rotating planes, 3 on each axis.

Not that I ever figured it out myself. In 7th grade, I got the book that showed you the solution and memorized it. Then I made $0.25 a cube by solving my classmates cubes. :)

Edited by Greebo
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  • 2 weeks later...

Seems like an enterprising young man! I just sold a speed cube for twenty bucks, and I'm hoping to sell 2 more.

On the math side of things, while it's true that you need know nothing of how the cube works, just a memorization/understanding of algorithms, I've long wondered how an understanding of the cube would affect my solves. I consider myself to have a pretty good understanding of how pieces are manipulated and I can look pretty far ahead when moving through certain parts of the CFOP system, such as the F2L and OLL. But how many more seconds does it take someone who can't look ahead and adjust?

Just some more food for thought.

Edited by Papes
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The number of unique positions is greatly reduced when you account for the fact if you have two positions that are the same except for having colors swapped, or simply having to rotate the whole cube, then they are really the same position.

I believe Rubik himself wrote a monograph on the mathematics of the cube.

Edited by punk
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