Rubik’s

Cube ME

Jane

Worley

Introduction:

Do

you ever think about all of the games that you played as a kid? Or puzzles that

tricked you and you could never figure out? The first thing that comes to my

mind when I think of troubling puzzles that were always difficult for me is a Rubik’s

cube. These are troubling puzzles for most people when they first pick them up,

after a little bit of practice some people can end up solving them, but others

cannot. The goal of the puzzle is to move all the edge cubies and corner cubies

into their correct positions with their correct orientation. This is a big

challenge for a lot of people because there is not a manual to the puzzle or a

YouTube tutorial video that tells you how to solve it.

This mind boggling puzzle was

created by a man named Erno Rubik who was a young professor of architecture in

Budapest in 1974. The object was supposed to not be possible to solve. His

creation was had a new type of thinking linked to it with its ability to twist

and turn and still not break and its colorful stickers on each side of the

cube. The cube took its own creator over a month to solve his invention, little

did he know at the time that his new invention and past time would turn into

one the world’s best-selling toys ever made. Erno was a teacher and was always

looking for new and innovative ways to teach his students by using techniques

that would help them better understand the material. He used the Cube’s first

model to explain to his students the concept of special relationships. Erno

thought of the Cube as an object of art, it was simple, but also complex. In

order for his brand new toy to become a best-seller, he had to put time and

effort into his work. It took showing his students his prototype for him to

realize the potential that was behind his creation. He got the Cube

manufactured, this took place in Hungary where it was made and

distributed. These first early cubes

were marketed as “Magic Cubes” and weighed about twice the weight as the ones

that are seen in stores today. During this time all Hungary was part of the

Communist regime and all imports and exports were closely watched and

restricted. So Erno had to figure out a way to get his product past Hungary and

into the hands of people all over the world. This was accomplished by

mathematicians and young entrepreneurs, they took them conferences and fairs

around their areas. At one of the events that it was taken to is where it was

discovered by a toy specialist, Tom Kremer, he agreed to sell it to sell it to

the rest of the world. Many names were thrown around that they thought would

work best to properly advertise this new and upcoming toy. It started out as

being called the “Magic Cube”, but investors thought that it related to

witchcraft so they changed it to “Rubik’s Cube”. And the infamous toy was

branded from there. Since the product was internationally launched in 1980

there have been an estimate of 350 million Rubik’s Cubes that have been sold

around the world. This cube that started out just as a tool for teaching

students soon became an iconic toy original to that decade. It started art

movements such as pop videos, Hollywood movies and even had its own TV show.

But it also came to represent both genius and confusion. The cube was widely

used all over the world and became such an iconic toy to those of all ages.

Rubik’s cube contains six center

pieces which do not move relative to each other, twelve edge “cubies” that each

have two stickers on them, and eight corner “cubies” that each have three

stickers on them. There is a typical cube solving procedure, first you need to

solve the two lower layers of the cube. Then orient the cubies in the top

layer, making sure that the top face of the cube is the correct color. And

finally you interchange the cubies in the top layer so that they are in their

correct positions. In order to properly solve the Rubik’s cube in a fast

manner, there are several shortcuts that you can memorize for dealing with

various cases and issues when it comes to the cube. The moves that one can

perform on Rubik’s cube form a mathematical structure called a group. And you

can solve the cube by using two basic ideas from group theory, they are

commutators and conjugation.

Investigation:

About 30 years after the Rubik’s

cube first appeared and became a well-known puzzle, an international team of

researchers proved that no matter how scrambled the cube got, it would always

be solvable. And to add to that statement, it could be solved in no more than

20 moves. But there was no way that these researchers could evaluate all of the

possible moves and cases that could appear when attempting to solve a Rubik’s

cube. Several of these researchers and professors came together and showed that

the maximum number moves required to solve a Rubik’s cube with N squares per

row is proportional to N^2/log N. The standard way to solve a Rubik’s cube is

to find a square that’s out of position and move it to the right place while

leaving the rest of the cube as little changed as possible.

The team of highly qualified

individuals recognized that under some events that a single sequence of twists

could move multiple squares into their proper places, this would potentially

cut down the total number of moves that it would take to solve the cube. There

is something called “God’s Number”, which is the 20. This is minimum amount of

moves that it would take to solve a Rubik’s Cube. It is called “God’s Number”

because it is very difficult to solve a Rubik’s Cube in general, but to solve

it in only 20 moves is pretty difficult and this means that any configuration

of a Rubik’s Cube can be solved in 20 moves or less. “There are about 43,252,003,274,489,856,000 possible

configurations when it comes to a Rubik’s Cube.”1 Each of

these configurations are classified by the minimum amount of moves that it

takes to solve the troublesome puzzle, this is called ‘distance’ of the

configuration.

If

you look at the table above, it shows that “more than two-thirds of the 43

quintillion possible configurations are at distance-18.”2

And then also a majority of the configurations fall into the categories of

distance-15 and 20. So if you pick up a random Rubik’s Cube that is completely

mixed up already, there is a higher chance of you getting a distance-18

configuration. So what are the odds that you will be able to solve the Rubik’s

Cube in 18 moves? In order to figure that question out you need to know how

many possible moves there are to solve the Rubik’s Cube.

There are six sides to the Rubik’s

Cube, each of these can be turned 1, 2, or 3 times. If they are turned four times,

then it just goes back to the original state it was in. So 6×3=18 which backs

up the fact of solving the Rubik’s Cube in 18 moves. So if there are 18 moves

made randomly, then the odds of solving the Rubik’s Cube is 1 in 18^18. This is

such a low percentage that someone cannot really solve a Rubik’s Cube on

accident. They would need to know patterns and formulas that would help them.

In something called statistical

physics there is a hypothesis that is called the ‘ergodic hypothesis’. “This

hypothesis says that any given configuration that is in statistical physics is

equally likely, and if you wait long enough then the system will eventually

pass through every possible state.”3

Now this hypothesis “would not be relevant to a Rubik’s cube where you use an

algorithm, but a Rubik’s cube making random moves is exactly the sort of

‘random walk’ that the ergodic hypothesis applies to.”4

1 “What

are the odds of solving a Rubik’s cube by making random moves?” Quarks &

Coffee, 13 Dec. 2015,

quarksandcoffee.com/index.php/2015/12/13/what-are-the-odds-of-solving-a-rubiks-cube-by-making-random-moves/.

2 “What are the odds

of solving a Rubik’s cube by making random moves?” Quarks & Coffee,

13 Dec. 2015,

quarksandcoffee.com/index.php/2015/12/13/what-are-the-odds-of-solving-a-rubiks-cube-by-making-random-moves/.

3 “What are the odds

of solving a Rubik’s cube by making random moves?” Quarks & Coffee,

13 Dec. 2015,

quarksandcoffee.com/index.php/2015/12/13/what-are-the-odds-of-solving-a-rubiks-cube-by-making-random-moves/.

4 “What are the odds

of solving a Rubik’s cube by making random moves?” Quarks & Coffee,

13 Dec. 2015,

quarksandcoffee.com/index.php/2015/12/13/what-are-the-odds-of-solving-a-rubiks-cube-by-making-random-moves/.