# Rubik’s over a month to solve his invention,

Rubik’s
Cube ME

Jane
Worley

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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/.