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Why Study Math? - How About Solving Rubik's Cube?

Joe Pagano

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So you think that math does not teach you things like solving Rubik's cube? Then think again. My ability to figure out the 3x3 cube lies not so much in my inherent brilliance but because my mathematical training taught me how to problem solve and to figure out things that without such training, I would never have figured out, or even attempted. Not that the Rubik's cube is as difficult as the Einstein Field Equations, but with 43 quintillion possibilities, it is certainly no walk in the park either. The wonderful field of mathematics helped train me to tackle this world-famous brain buster.

The Rubik's cube is actually a very interesting mathematical object. Its mathematics lies within the realm of group theory and the cube itself is known as a permutation group. Specifically, the cube is a group which is generated by the rotations of its six faces. The size of this group is enormous-to the tune of 43 quintillion (a quintillion is a 1 followed by 18 0's-for more on this, see my article “Numbers - How Big is Big?"), and the set of permutations generated by the cube is governed by typical group properties. Let me explain.

A group is a set of objects that has the property that when you perform the mathematical operation imposed on the group, you end up with another object within the set. The “mathematical operation" associated with the Rubik's cube group is the rotation of its faces. Each of these moves is known as a permutation because we are permuting the color facelets on the cube, but we are still ending up with a cube, albeit in a different arrangement of the color facelets. This group operation of “rotation of the faces of the cube" must also be associative, which in this case it is, since we can take three moves of the cube, call them A, B, and C and associate them as (AB)C or A(BC) and end up with the same ultimate move. Each move has an opposite, or inverse (just reverse the rotation); moreover, though not obvious, the cube has an identity element which will return any position of the cube, or any arrangement of the cube, back to its original position. The existence of these properties impose a group structure on the Rubik's cube.

Now anyone can cheat and learn to solve the cube by studying another's solution. The trick is to solve it using your own solution-that is figuring out the cube. The cube can be very frustrating but what if you were stranded like Tom Hanks in the movie Castaway? Do you think you would figure it out given enough time? Or would you do what most do and quit? What if someone said that in order to get off that tropical island you had to figure it out? Faced with this ultimatum, I can pretty much say that you would come up with some creative problem solving strategies to solve that cube and get off that island. Unless of course, you like coconuts and crab that much.

Now what you would probably do in this situation, after much trial and error, is to find out how to get the cube solved to the top layer. You would get to this point after much practice; after many, many mistakes, false starts, and restarts; and after much head banging. You would then ultimately figure out how to get the remaining 9 facelets into their correct positions and orientations, discovering certain moves which fixed certain cubes and changed others more to your liking.

After much wrangling and brain-sweating, you would come to know the moves that oriented your facelets in exactly the way you needed them when you got to that top layer. In fact, these “orientation moves" are actually permutations which preserve certain positions of the cube while altering others so that the cube can be restored to normal, normal being the solved position, or the one in 43 quintillion!

The point of this article is that you have the power to solve the cube if you put your mind to it. And what a sense of satisfaction you will get when you know you just whipped into shape a permutation group of size 43 quintillion! So try the cube and remember: you can solve it if you only try. I did it and so can you.

See a list of my ebooks here at Order Ebooks

Joe is a prolific writer of self-help and educational material and an award-winning former teacher of both college and high school mathematics. Joe is the creator of the Wiz Kid series of math ebooks, Arithmetic Magic, the little classic on the ABC's of arithmetic, the original collection of poetry, Poems for the Mathematically Insecure, and the short but highly effective fraction troubleshooter Fractions for the Faint of Heart. The diverse genre of his writings (novel, short story, essay, script, and poetry)-particularly in regard to its educational flavor- continues to captivate readers and to earn him recognition.

Joe propagates his teaching philosophy through his articles and books and is dedicated to helping educate children living in impoverished countries. Toward this end, he donates a portion of the proceeds from the sale of every ebook. For more information go to


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