Solving the Rubik s Cube The Math Behind the Cube: How many different combinations are possible on a 3x3 cube? There are 6 sides each with 9 squares giving 54 squares. Thus there will be 54 53 52 51 50 49... 5 4 3 2 1 or approximately 230 843 697 300 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 combinations This number is unrealistic as we need to think in terms of number of smaller cubes not stickers that make up the cube (each corner piece has 3 faces, and these 3 faces make up 1 piece). So there are 26 smaller pieces that make up the large 3x3 3 cube. (The middle cube doesn t count 3 1 26.) Thus there will be 26 25 24 23 22 21... 5 4 3 2 1 or approximately 403 291 461 100 000 000 000 000 000 combinations
However, the middle of each face is connected to a central core. In other words the middle piece of each face are all connected together and will not move in relation to each other. Blue is always opposite Green, Red is always opposite Orange. Yellow is always opposite White. Therefore these middle pieces all count as one piece not 6 separate pieces giving us 21 pieces total (8 corners, 12 edges, and 1 central core). Thus there will be 21 20 19 18 17 16... 5 4 3 2 1 or approximately 51 090 942 170 000 000 000 combinations As we begin solving a cube you will begin to notice that some states are impossible to achieve. Typically you can never affect only one piece of the cube but instead will affect 3 pieces at once. In fact the number of pieces affected by a move or a series of moves will be a number divisible by 3. Due to this fact the above number is still a bit too high. The actual number of combinations is 43 252 003 274 489 856 000. All we need to do is find the one that works!
How to Reference and Talk About the Cube: As we have already discussed the center pieces are all connected to a central core and do not move in relation to each other. Blue is always opposite green, red is always opposite orange, and yellow is always opposite white. When solving, we will then place the other pieces around this central core where they belong. Labeling the Cube: Up Face U Back Face B Left Face L Right Face R Front Face F Down Face D
Movements on the cube: To make life easy we will use a notation involving U, D, R, L, F, B to signify which face will be turned. We either turn a face clockwise or counterclockwise. To decide which way to turn a face imagine you are looking directly at the face when turning. Ex) R means turn the Right face clockwise R means turn the Right face counterclockwise *again to help determine which way is clockwise and which way is counterclockwise imagine you are looking directly at the Right face.
Building the First Side
Placing Corners Position the corner to be placed directly below the space it is to occupy. This is called having the same relative position. Swing the desired piece out of the way. D D Bring down the face containing the space you wish to occupy. R L Swing the desired piece into place. D D Restore the top face now with the corner in place. R L
Placing Centers Position the edge to be placed so that it has the same relative position of the space it is to occupy. Swing the desired piece out of the way. U D Then reface cube U D Then reface cube Bring down the face containing the space you wish to occupy. R L Swing the desired piece into place. U D Then reface cube U D Then reface cube Restore the top face, now with the desired edge in place. R L
Bringing Pieces Into Play Corners on Top Facing the Side Position the cube so the desired color of the corner is facing you. Turn the front side down so the desired face is now on the bottom layer of the cube. F Swing the desired piece out of the way. F B Restore the top side of the cube. B F F
Corners on Bottom Facing Down Position piece to move beneath a corner that is not yet finished. R Rotate the face containing both the desired piece and the empty corner so that the desired piece is on the bottom of the cube with the desired color now facing a side. Swing the desired piece out of the way. B Restore the top face of the cube. R
Edges on Top Facing the Side This will bring an edge on the top layer of the cube that is not facing the right way into the middle layer of the cube so that it can be worked with. Bring down the face containing the edge you would like to move. F Swing the edge you want moved out of the way. U D Then reface cube Restore the top face of the cube. F
Edges on Bottom Facing the Side Position piece to move beneath an edge that is not yet finished. Rotate the face containing both the piece you want to move and the empty edge so that both are in the middle layer of the cube. F Swing the edge you want to move out of the way. U D Then reface cube Restore the top face of the cube. F
Edges on Bottom Facing Down Position piece to move beneath an edge that is not yet finished. Rotate the face containing both the piece you want to move and the empty edge so that both are in the middle layer of the cube. F Swing the edge you want to move out of the way. U D Restore the top face of the cube. F
Building the 2 nd Layer
We will now move on to the second or middle layer of the cube. To complete this stage and in fact to complete the rest of the cube, you will need to hold your cube with its completed side down. This stage of the solution involves only one series of steps or algorithm as we only need to place the four edge pieces. To begin with rotate your finished face so that the side colors match the center colors found on the four faces coming up from the finished side as shown below. will move piece down to the left U Situate the desired edge piece found on the top layer so that the side color facing you matches the middle color facing you Move the desired piece to the opposite side of where it needs to end up will move piece down to the right U
L U L Bring up the face that contains the location of where you want to end up. Bring the desired edge piece back to the front so its side color once again matches the middle color facing you. Bring the side face back down so that the desired edge piece now matches the corner piece. (These two pieces will now be kept together as a pair.) Swing the new edge / corner pair out of the way. R U R U Rotate the front face to bring up the corner that has been removed. U F F
Swing the matched edge / corner piece pair back to the front. U Rotate the front face back down bringing the edge / corner piece pair into their proper place. U F Summary of the Algorithm (Moving the piece down and to the left) U L U L U F U F F Summary of the Algorithm (Moving the piece down and to the right) U R U R U F U F
Finishing the Top Face
Part I Flipping the Edges Up In this step we will bring all edge pieces up so that their color corresponds to the top center color. We will worry about putting them into their proper locations later. At this stage your cube s top face will have no edge pieces facing up, two edge pieces facing up that are across from each other, two edge pieces facing up that are next to each other, or all four edge pieces facing up. (It is impossible to have just one or three edge pieces facing up.) Top Face No edges up 2 edges up 2 edges up All edge up across from next to each (Stage I finished) each other other The same move will be used to bring the cube through these various states. Simply start on whatever state your cube is currently at.
Begin each algorithm (series of movements) holding the cube as shown below. Will become Simply perform the algorithm R U F U F R Will become to go from one state the next. Will become
Part II Flipping the Corners Up As stated earlier, we always affect three pieces of the cube at a time. For this move to be successful, we need to begin with one corner facing up and three facing the sides. If your cube is not in this state we need to first perform the algorithm to get there. At most you will need to do the complete algorithm twice. Flipping the Left Side Flipping the Right Side Top right corner is unaffected. Top left corner is unaffected. L U L U L U(2) L R U R U R U(2) R
When your cube has one corner facing up and three corners facing the sides it will look like one of the following states. Performing the algorithm below will complete the stage. Performing the algorithm below will complete the stage. L U L U L U(2) L R U R U R U(2) R
Part III Placing the Corners We will now place the corners into their proper locations. Again we will affect three corners at a time so we will begin this move with exactly one corner in its proper location. Rotate the top face so this is the case. If this is not possible, perform the algorithm given below with no corners in their proper location. When finished, you will be able to create the scenario needed. Hold your cube so that the corner that is correct is on the front face in the top left position. R F R B(2) R F R B(2) R(2) If at first the algorithm does not move all corners to their proper location simply repeat it. Remember, in order to work only one corner can be in its proper location. Place this corner at the as shown above, then perform the algorithm.
Part IV Placing the Edges The last stage places the edges into their proper locations. Chances are one edge will already be in its proper location leaving three to be placed. (Again with the three!) Hold your cube so the correctly placed edge is at the back of the cube. Rotates pieces counterclockwise. Start on left side. L U L U L U(2) L Rotates pieces counterclockwise. Start on left side. R U R U R U(2) R Now hold the cube like so and restore the corners. Now hold the cube like so and restore the corners. R U R U R U(2) R Finished!!!!!!! L U L U L U(2) L