“Our greatest weakness lies in giving up. The most certain way to succeed is always to try just one more time” (Thomas Edison)
The Rubik’s cube, first invited by Mr Ernő Rubik in 1974, is one of the most wellknown puzzles today, and I’m sure the majority of people will have one in their home, either unsolved and gathering dust on a shelf or in the attic, or solved due to much thought and much labour (or the cheap trick of peeling off the stickers).
At the age of 8, I became very interested in this 3D combination puzzle of the 1980s and spent many hours working out how to solve it. However, since then I have managed to solve it numerous times, using a set of steps starting by solving the first face with all pieces orientated in the correct manner, and then working my way up to the top layer. Nowadays I have an average of a 45 second solve, for the regular 3x3x3 Rubik’s cube, which sounds great, until you take a look at the world record which accounts to an outstanding 5.55 seconds (by Mats Valk from the Netherlands).
However, without a photographic memory, getting ridiculously short times proved to be challenging and soon the regular 3x3x3 became relatively dull to me. However, I managed to satisfy my appetite through a variety of other 3D puzzles, all of which were spawned from the one invention of Mr Rubik. Cubes today range from 8x8x8 Rubik’s cubes to ones with shapeshifting qualities. The principles of the 3x3x3 can be applied to the majority of these other cubes which led me, like many others before me, to solve these impeccable puzzles.
What I find
most interesting, though, about the Rubik’s cube is the mathematics
behind it.
Being a person who enjoys a good bit of maths, naturally that would arouse my curiosity. People always find it hard to solve the Rubik’s cube and many people just keep turning the layers, hoping that they will eventually fit into place. I then wondered: what is the possibility of solving the Rubik’s cube, purely by chance alone? The first thing I then did was work out how many possible orientations of the Rubik’s cube are available. Any of the 8 corner pieces can reside in one of the corner slots, therefore 7 corner pieces could reside in one of the other corner slots, 6 corner pieces for the next corner slot, and so on. Therefore for the positioning of the corner pieces you get 8x7x6x5x4x3x2x1= 8! Each corner piece can then go into each slot in 3 different ways (as each corner piece consists of three colours), as there are 8 different corner pieces we get 3^{8} different ways of orientating each corner piece. If you then apply the similar method of working to the edge pieces (N.B the centres can’t move so they aren’t necessary), you can see that there are 12! possible positioning’s as each edge piece can be placed in 12 different places. Moreover there are only 2 possible orientations of the edge piece giving 2^{12} possible orientations. If you then times all this together, 3^{8}2^{12}8!12!, it gives one 5.19 x 10^{20} (519 quintillion) possible configurations of the Rubik’s cube. Phew! That is indeed a very large number of possible configurations, so you can already see, that solving this without any method or help will prove extremely challenging, however not all hope is lost for not all those configurations are physically possible due to the nature of the cube. The total number of states is actually yielded by a set of values obtained by a robot programmed to execute a set of random 90 degree turns on a regular Rubik’s cube. What this resulted in was a multiplication of 2^{27} x 3^{14} x 5^{3} x 7^{2} x 11, which gives a final answer to the total number of cube positioning as 43, 252, 003, 274, 489, 856, 000 or 43 quintillion.
Being a person who enjoys a good bit of maths, naturally that would arouse my curiosity. People always find it hard to solve the Rubik’s cube and many people just keep turning the layers, hoping that they will eventually fit into place. I then wondered: what is the possibility of solving the Rubik’s cube, purely by chance alone? The first thing I then did was work out how many possible orientations of the Rubik’s cube are available. Any of the 8 corner pieces can reside in one of the corner slots, therefore 7 corner pieces could reside in one of the other corner slots, 6 corner pieces for the next corner slot, and so on. Therefore for the positioning of the corner pieces you get 8x7x6x5x4x3x2x1= 8! Each corner piece can then go into each slot in 3 different ways (as each corner piece consists of three colours), as there are 8 different corner pieces we get 3^{8} different ways of orientating each corner piece. If you then apply the similar method of working to the edge pieces (N.B the centres can’t move so they aren’t necessary), you can see that there are 12! possible positioning’s as each edge piece can be placed in 12 different places. Moreover there are only 2 possible orientations of the edge piece giving 2^{12} possible orientations. If you then times all this together, 3^{8}2^{12}8!12!, it gives one 5.19 x 10^{20} (519 quintillion) possible configurations of the Rubik’s cube. Phew! That is indeed a very large number of possible configurations, so you can already see, that solving this without any method or help will prove extremely challenging, however not all hope is lost for not all those configurations are physically possible due to the nature of the cube. The total number of states is actually yielded by a set of values obtained by a robot programmed to execute a set of random 90 degree turns on a regular Rubik’s cube. What this resulted in was a multiplication of 2^{27} x 3^{14} x 5^{3} x 7^{2} x 11, which gives a final answer to the total number of cube positioning as 43, 252, 003, 274, 489, 856, 000 or 43 quintillion.
One could
also bring in the second law of thermodynamics to help show the improbability
of obtaining a solved cube, although this is theoretical and the reality is not
the case. The law states that in a thermodynamic process, there is in an
increase in the sum of the entropies of the participating systems. In terms
most people are more likely to understand; a system tends to move from a state
of order to a state of disorder. So this could be the scrambling of the cube,
it goes from a state of order, solved with all pieces in their correct places,
to a state of disorder, where all the pieces are in a completely jumbled up
state. So this makes it very hard to solve the Rubik’s cube without external
thought around where the pieces are supposed to go, and thought around
algorithms to place the pieces in their correct slots.
Mentality is
one of the other problems people face when trying to solve the Rubik’s cube, if
you don’t have a desire to solve it, you are unlikely to solve it, and it is
one of those things which takes time and effort. Most people don’t have the
patience to solve the Rubik’s cube and thus it gets strewn across the room in a
fit of rage and anger. What people also don’t understand is that there is a
logic behind solving the Rubik’s cube, it cannot be done by simply twisting the
layers round and round. One of the other things people ask me about the Rubik’s
cube, is “is there a set of moves which you repeat and will always get the cube
into the right place?” well no you cannot do just a string of 4 moves over and
over to solve the cube for the reason I mentioned earlier, there is about 43
quintillion different ways the pieces can be represented, this means one cannot
simply do one algorithm over and over.
I would love to give a full tutorial right here about how the cube can be solved, however it will take too long to transmit to words. However I can give you a brief overview to get you on your way.
Start by picking a face to start solving, for example white. Find the white centre of on the cube, and form a white cross around that centre with edge pieces, this can be done by a few simple 90degree turn moves. Once you have the cross, check the orientation of your edge pieces. Is the white and red edge piece in line with the white centre and the red centre, if not make a 180 degree turn to put that piece on the top layer and turn the top face round to line up the red, and make another 180 degree turn to put it in place. That’s the easy bit. Now work on the corners get them in the right places again, by simple 90 degree turns of the top face and the side faces. Right, now your own your way. The real key to solving the Rubik’s cube though is understand what each turn move does and develop from that, ways of moving pieces around the cube, while making sure other pieces move back into their original positioning. Or if you’re really desperate I am sure there are many a video on YouTube describing how it can be solved.
All of these things make a very challenging
puzzle but recent research has made an amazing discovery. Tomas Rokicki,
Herbert Kociemba, Morley Davidson, and John Dethridge, proved that a regular
3x3x3 Rubik’s cube can be solved in 20 moves or less from any start position.
This is incredible news and makes the number of moves I take look stupid, for
on average I take about 105, 90degree turns of the cube in one solve,
considerably more than 20. However to this you will have to memorize the
algorithms for each of the 43 quintillion different possibilities of starting
positons, as appose to my method of solving layer by layer. The way that this
was discovered by that group of people, was through a branch of mathematics
known as group theory. I would like to go into the gory details, but it would
require me to give a vast amount of detail into notation and theory, which
would take far too long, however look online for a paper on Group Theory and
the Rubik’s Cube if you are interested, it is written by Janet Chen. Back to
how it was done. They first split the 43 quintillion into 2,217,093,120 sets of
19,508,428,800 different positions. Each set was able to be worked out using a computer,
and a variety of Cosets of the group formed by {U,F2,R2,D,B2,L2}, where U is
the upper layer of the cube, F2 is the front face, R2 is the right face, D is
the bottom face, B2 is the back face, and L2 is the left face. The number of
sets needed to be solved was cut down to 55,882,296 due to symmetry around the
cube. They then used a number of superfast computers to rapidly go through each
set and calculate the shortest number of moves to solve it. They then used this
to prove the number of moves it takes to solve the cube from each start
position. The results are represented in the table below. Where distance is the
number of moves from the start position. I would love to give a full tutorial right here about how the cube can be solved, however it will take too long to transmit to words. However I can give you a brief overview to get you on your way.
Start by picking a face to start solving, for example white. Find the white centre of on the cube, and form a white cross around that centre with edge pieces, this can be done by a few simple 90degree turn moves. Once you have the cross, check the orientation of your edge pieces. Is the white and red edge piece in line with the white centre and the red centre, if not make a 180 degree turn to put that piece on the top layer and turn the top face round to line up the red, and make another 180 degree turn to put it in place. That’s the easy bit. Now work on the corners get them in the right places again, by simple 90 degree turns of the top face and the side faces. Right, now your own your way. The real key to solving the Rubik’s cube though is understand what each turn move does and develop from that, ways of moving pieces around the cube, while making sure other pieces move back into their original positioning. Or if you’re really desperate I am sure there are many a video on YouTube describing how it can be solved.
The number of moves to solve the Rubik cube has also be represented by Erik Demaine as where n is the number of layers in a (n*n*n) Rubik’s cube, and is a constant.
As you can
see the Rubik’s cube requires a lot of thought and patience in order to
complete, however it can be done and has on many occasions been done by your
average Joe, such as myself. However when you delve into the maths behind the
Rubik’s cube things become even more complicated. And to motivate you in all
that you do, be the Rubik’s cube, an area of your course you find hard, a
musical instrument or anything.
Distance

Count of Positions

0

1

1

18

2

243

3

3,240

4

43,239

5

574,908

6

7,618,438

7

100,803,036

8

1,332,343,288

9

17,596,479,795

10

232,248,063,316

11

3,063,288,809,012

12

40,374,425,656,248

13

531,653,418,284,628

14

6,989,320,578,825,358

15

91,365,146,187,124,313

16

about 1,100,000,000,000,000,000

17

about 12,000,000,000,000,000,000

18

about 29,000,000,000,000,000,000

19

about 1,500,000,000,000,000,000

20

about 490,000,000

wow
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