# The Cube in computer science, and different ways

The Rubik’s cube is a puzzle in the form of a plastic cube covered with multicolored squares, which the player attempts to twist and turn so that all the squares on each of the 6 faces are of the same color. Erno Rubik’s cube was created as a teaching tool many years ago. It then quickly became a popular toy. This is evident due to the fact that people of all ages have spent weeks upon weeks trying to solve the Rubik’s Cube, and professors have written paper dedicated to just the math behind the Rubik’s Cube. The cube, made up of nine colored squares on each side, can be rearranged over 43 quintillion different ways. The exact number is 43,252,003,274,489,856,000.            The Rubik’s cube can be so mind boggling that after Erno Rubik’s designed the “magic cube” as he called it, he realized he could not actually solve the puzzle.

This paper will talk about the history of the Rubik’s Cube, the mechanics of the Rubik’s Cube, how it can be used as a teaching tool (algorithms), Rubik’s Cube notation, Rubik’s Cube in computer science, and different ways to solve the Rubik’s Cube.The History of the Rubik’s Cube                                                                                                                                                The Rubik’s Cube was created in 1974 when Hungarian architect Erno Rubik wanted a working model to help explain three-dimensional geometry. After designing it, he realized he could not actually solve the puzzle. The more he moved the colored squares, the more mixed up they became. “It was a code I myself had invented!” he wrote.

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“Yet I could not read it.” Erno Rubik took a couple years before taking his invention to the toy market, and in 1979 he presented it at Nuremberg Toy Fair where it was seen as a possible success. The Rubik’s Cube got a licensing deal to the Ideal Toy Corp in 1980 and, by January 2009, more than 350 million units had been sold worldwide, making it the biggest-selling toy of all time. It has decreased in popularity since but it still never fully disappeared due to the hobby of speed cubing, when people try to solve The Rubik’s Cube really fast.The Mechanics of the Rubik’s cube                                                                                                                                                            The Rubik’s Cube is consisted of six colored faces. Each face has a total of nine tiles making fifty-four altogether.

These colored tiles are grouped into twenty-seven pieces. There are eight corners, twelve edges, six centers, and one cube in the middle (it cannot be seen). The centers never move; they only rotate in place. No moves can ever make an edge be a center or a corner. An edge has 2 colors on it, which also never change.

No moves can ever make a corner be a center or an edge. A corner has 3 colors on it, that never change. This means that not all color patterns are possible. For example, it is impossible for a corner that has a green tile and for a corner that has a blue tile to be on the same face, since they are opposite sides.

There are few ways to move the pieces of a Rubik’s Cube. The only moves you can make is to rotate a face. There are only 3 such rotations possible for each face: rotate 90 degrees clockwise, rotate 90 degrees counter-clockwise, or rotate 180 degrees.

How Can a Rubik’s Cube be used as a teaching tool?             The Rubik’s Cube can be used to teach basic geometry. Students may learn about    3-D geometric shapes and the behavior of them. For example, The Rubik’s Cube is a cube. It has six faces, 12 edges, and eight vertices. As it is known as a 3x3x3 cube a student can be shown how volume works. For example, if the cube was filled with cubes it could fit twenty-seven of them.

Another way the Rubik’s Cube can be used as a teaching tool is that it can help teach the surface area of a cube.  An example is that the three squares from one face multiplied by another three square equals the nine total squares that can be counted on an individual face.Rubik’s Cube Notation             Basic Rubik’s Cube notation is fairly simple. The six faces of a Rubik’s Cube in notation are named right (R), left (L), up (U), down (D), front (F), and back (B). If you wanted to move a face counterclockwise, you would indicate that by writing one of these letters. While if you wanted to move one of the faces counterclockwise you would add a lowercase “i” to indicate that you are inverting the face. Also if you want to move a side twice you would add a 2 after the letter.

For example, a notation might be R U R’ U R U2 R’ U. To name a corner cube of The Rubik’s Cube you simply name the visible face color from that cube in clockwise order. For instance, the upper, left, forward corner is written as the ulf corner. It can also be called lfu or ful corner.

Besides whole face rotations there are also slice turns, double layer turns, inverse double layer turns, and whole cube rotations that can be written in notation form. The letters used for slice turns are E, M, and S. E stands for the equator or the layer between U and D, it this notation indicates that you are turning the middle layer toward D. M indicates the movement of the middle layer or the layer in between L and R turning into the direction of L. Finally, there is S which indicates the turning of the layer between F and B, turning the layer toward F. To invert all the direction, you add a lowercase “i” just as you would for face rotations. Double layer turns are treated just like regular face turns except the letters are lower case to how that two layers are being turned. Likewise, to invert the double layer turns a lowercase “i” is used.

Lastly there are whole cube rotations, these are used to change the orientation of the faces. The letters used are X for rotating the whole cube on the R, Y for rotating the cube on the U, and Z for rotating the entire cube on the F.Rubik’s Cube in Computer Science             Computers can be used to help devise algorithms for solving and making different patterns on the Rubik’s Cube. Computers can also be used to memorize long and complicated algorithms that are near impossible to memorize such as Thistlethwaite’s 52-move algorithm which is such a concise and complicated method it cannot be physically memorized by the average innocent cuber. Computer programs can also be downloaded into man made machines to scan the colors of mixed up cubes and devise an algorithm and then solve the cube right there and then. Such machines exist all over the world, one being at the University of Sheffield.

The robot called the Sub1 Reloaded can unscramble a Rubik’s Cube in 637 milliseconds – considerably less than the fastest human time of 4.9 seconds. This lightning quick time will go in The Guinness World Record for the fastest time that a robot had solved a cube in.  Another example of Rubik’s Cubes being solved by robots is at the Beyond Rubik’s Cube exhibit at the liberty Science Center. The machine over there takes its time scanning the six faces of the Rubik’s Cube and solves it with ease in front of others. Different ways to solve the Rubik’s Cube             The regular 3x3x3 Rubik’s Cube can be arranged over 43 quintillion different ways. It is not clear exactly how many algorithms there are that can be derived from that number of arrangements, but it is surely a vast number.

Nevertheless, scientists have used Googles super computers to figure out what the maximum necessary amount of moves are needed to solve a Rubik’s Cube. The answer that they found was twenty. Only twenty moves are needed at most to solve a regular 3x3x3 Rubik’s Cube. Still there are an abundance of possible algorithms that are used to aid the solving of a Rubik’s Cube, thankfully there are efficient, simple, and well known algorithms that can help solve a Rubik’s Cube.

One of the easiest beginner algorithms is commonly known as the layer method. This is when the person solving the cube takes it layer by layer until the entire cube is finished. Algorithms are generally only memorized for the middle and last layers, which is why this is the beginner’s choice. Another basic method which is known as the Corners First Method calls for the solver to align all corners into their proper positions and then fill in the edges. A more intermediate method would be the Fridich Method which is by far the most popular method for speed cubers. First the bottom layer edges are solved, then the first two layers are filled in either using intuition or algorithms, and finally the top layer is solved in two steps. Finally, a very advanced method of solving the Rubik’s Cube is the Human Thistlethwaite method.

It is a human-usable version of the Thistlethwaite algorithm that reduces the cube to subgroups, and ends by solving the cube with only 180-degree face turns.Here’s an example of how the algorithms from the layer method are applied to the Rubik’s Cube. The first step in this method is to solve the cross on the top face, generally people make the top face the white side. However, the algorithms work no matter what side you start on. The top cross consists of only four edge pieces and one center piece these pieces have to be oriented in correct position in retrospect to the colors of all the other faces.

Therefore, the algorithms used are applied only to specific cases. For example, if three edges are done and the last one is oriented wrong the algorithm used is F R’ D’ R F’ F”. Another case is if last white edge is on the front face but it can’t be moved to the top because it’s oriented wrong the algorithm used for this case is F’ R’ D’ R F’ F’. These two algorithms will solve the cross and once the cross is solved the next step is to solve the corners of the first layer. There are three possible cases for where the white corners are located. As a result, there are three algorithms that are used. The first algorithm is used when the white corner is in correct orientation but it is in the bottom layer. The algorithm used is R’ D’ R.

A second case is when the white piece is on the opposite side of the cube and in the bottom layer. The algorithm used for this case is F D F’. The final case is when the white corner piece is in the top layer just in a wrong spot or oriented bad. You need to take the corner out of the top layer and move it to the bottom layer and then you use one of the previous two algorithms.

This algorithm is R’ D2 R D R’ D’ R, and then either R’ D’ R or F D F’.As we move on to the next step the algorithms get more complicated. This is because the second layer has to be solved without messing up any pieces in the first layer.  When solving the second layer there are two cases which someone may face. Either the cube is in the bottom layer or it is already in the second layer but just in the wrong spot. When moving a cube form the bottom layer to the second layer there are two possible algorithms that are used.

The purpose of these algorithms is to bring the front-up edge to the right-front or left-front positions. It should be noted that the cube is now flipped so that the solved side is facing downwards. The first algorithm that can be used brings the front-edge to the right- front, the algorithm is U R U’ R’ U’ F’ U F.

The second algorithm bring the front-edge to the left-front, this algorithm is U’ L’ U L U F U’ F’. The final algorithm is used when the edge piece is in the second layer but is oriented wrong. The algorithm used is U R U’ R’ U’ F’ U F – U2 – U R U’ R’ U’ F’ U F. Note that the second layer is solved the next step is to solve the final layer. The first part of this process is to once again solve the top cross. The algorithm used here is fairly simple and it is used either once or twice depending on the orientation of the cube. The algorithm is F R U R’ U’ F’ and it is used twice if the top cubes are in the shape on an “L”, it is used once if the cubes are in a straight line.

After the top cross pieces are put into place they then have to be oriented correctly so that they are in line with the other colors of the center pieces. The algorithm used here is R U R’ U R U2 R’ U and the purpose of this algorithm is to switch the pieces so they are in the correct place. Again the algorithm is used either once or twice until the cubes are in correct position. After this the next step is to position the yellow corners into the right spots according to the color.  It does not matter yet that the cubes are in the wrong orientation.

The algorithm used for this section is U R U’ L’ U R’ U’ L. To execute this algorithm, you have to first find a yellow corner which is on the right position, then place this one in the front-right-top of the puzzle and execute the permutation. If the pieces didn’t get where they belong the algorithm is used one more time. The final step used after the corners are placed in the right position is to orient them correctly. There is only one algorithm for this step and it is R’ D’ R D. However, this algorithm is repeated multiple times in a methodical fashion.

You have to hold the cube in your hand so that the upper piece you want to orient is on the front-right-top corner, then do the R’ D’ R D algorithm twice or four times until that specific piece is oriented correctly. After you orient one piece you use a U’ to get to another piece that is oriented incorrectly again you repeat the R’ D’ R D algorithm until it is oriented correctly. This process is used until all pieces are oriented correctly if done correctly the Rubik’s Cube should now be solved.             The Rubik’s Cube is a simple complexity that boggles the mind of adults and children alike. Its simple structure makes it seem like solving it would be an easy task however one may find that the more you turn the layers and faces, the more mixed up it gets. There are many uses for the Rubik’s Cube and secrets behind it. It can be used as a teaching tool for basic geometry and it is the bearer of a code of letters known as cube notation.

The Rubik’s Cube can be applied in computer science, just to derive combos trying to figure out how to solve it. Overall it truly is a wonder.

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