Patent filed by Erno Rubik 1975, sold by Ideal Toys in the 1980's.

(plastic with colored stickers, 2.2"; keychain 1.2")

The first puzzle of this type in a large class of puzzles in the years to follow. Challenging and fun to play with. A number of ways to construct this puzzle have been devised over the years; here are the pieces of an original Rubik's Cube like shown above, where there is a central axis assembly and 20 pieces that interlock with it.

- Solve the top layer (all of it, including the sides), and turn the cube over so now it the bottom layer and the bottom third of the cube is solved (easy with a little practice).

- Solve the middle layer:

Rotate the middle so centers are correct, and then move edges between the up and middle layers until the middle is solved. If an edge first needs to be flipped, move it be FU and do the edge flipper sequence of Step 3 (the edge be flipped, and you can rotate the top to move it back to be FU). Parentheses are just to make the sequence easier to read.edge mover, FU -> FR:(U R) (U- R-) (U- F-) (U F)- Flip the up edges so they all have the correct color on top:

If no up edges have correct top color, first do the edge flipper. Now position the cube so UL has correct top color and UF does not, and do the edge flipper at most two times.edge flipper:F (R U) (R- U-) F-- Move the up layer edges to their correct positions:

As needed, re-position the cube and use the edge swapper sequence.edge swapper, UF<->UL:(R U) (R- U) (R U^{2}) (R- U)- Position the up layer corners:

The corner cycle sequence leaves UFR alone and cycles the other three counterclockwise. Identify one corner that is correct (but may be rotated), or if there is not one, do the corner cycle. Then re-position the cube so the correct corner is UFR, and then do the corner cycle one or two times to make all corners correct.corner cycle:(U R) (U- L- ) (U R-) (U- L)- Rotate the up layer corners (read this whole step before starting it):

*** Don't worry that the bottom is mixed up as you do this, it will be ok in the end.

Position the cube so UFR is not correct and repeat these two steps until all corners correct:

- Repeat the corner rotator until the UFR corner is correct:

corner rotator:R- D- R D

- Rotate the up layer (
not the whole cube) so that UFR is incorrect.

Each sequence has a natural rhythm, but an easy mistake is to start off wrong. The Edge Mover and Corner Cycle start withU, the Edge Flipper (after parking theF) and the Edge Swapper start withR. To avoid forgetting your place, run the sequence in your head, and when you get faster, simply count 1,2,3,4,... as you go; 8 for the edge mover, edge swapper, and corner cycle; 4 between the F's of the edge flipper; 2 sets of 4 for the corner rotator.

Edge Mover (for Step 2):edge mover, FU -> FR:(U R) (U- R-) (U- F-) (U F)

It starts with aU, and every other move involves aUorU-.

First two moves and last two moves are clockwise, middle four moves are counter clockwise.

First 4 moves involveR, second 4 moves involveF.Edge Flipper (for Steps 2 and 3):edge flipper:F (R U) (R- U-) F-

"Park" the front withF, do(RU) (R-U), and then "unpark" the front withF-.Edge Swapper (for Step 4):edge swapper, UF<->UL:(R U) (R- U) (R U^{2}) (R- U)

It'sR R- R R-interleaved withU U U. The^{2}UR's alternate + and -, and theU's keep going clockwise, where the third is 180 degrees.Corner Cycle (for Step 5):corner cycle:(U R) (U- L- ) (U R-) (U- L)

It'sU U- U U-interleaved withR L- R- L.Corner Rotator (for Step 6):corner rotator:R- D- R D

Always complete this sequence before doing Step 6B; it is easy to forget the finalDwhen you see the correct color on top.

It will be done twice (eight moves to rotate once) or 4 times (16 moves to rotate twice).

Step 1:After getting faster at the other steps, this step can become the slowest. Starting at one corner and working across the top works, but at each step one is hunting for one or two specific pieces to place next, and that can be slow. A faster approach may be to pick a middle piece, say white, and start with it on top. Then repeatedly look for the first white edge you can find and place it. Then repeatedly look for the first white corner you can find and place it. Although not necessary once you get fast, it can help to keep the middle layer aligned so each time you find a piece it is easy to see where it needs to go.

Step 2:Instead of using the edge flipper, learn the symmetric sequence that moves an edge down counterclockwise from up to middle:Or, use these two more complicated sequences that use only 7 moves:edge cc-mover, UF -> LF:(U- L-) (U L) (U F) (U- F-)alternate edge mover, FU -> FR:(L F^{2}) U F U- (F^{2}L-)

alternate edge cc-mover, UF -> LF:(R- F^{2}) U- F- U (F^{2}R)Step 3:Before the finalF-, if the right side of FR is not the top color, instead of wasting time to doF- F, repeat the(R U) (R- U-)before doingF-.

Step 4:If more than two edges need to be exchanged, rotate the top layer so the UF edge is correct and just do the first 7 steps of the edge swapper. Omitting the last move of the Edge Swapper leaves UF unchanged and cycles the other three counterclockwise. So you are done if that was needed. Otherwise, it turns out that now a second edge swapper operation, possibly followed by aU-, will always suffice. However, if you are willing to remember yet another sequence, a second edge swapper operation can be avoided for the case that a clockwise cycle is needed by using this sequence:clockwise cycle UL, UB, UR:(R U^{2}) (R- U-) (R U-) R-Step 5:If no corners are correct, learn how to tell for which orientation of the cube the corner cycle will leave things so that a counterclockwise cycle will be needed. Or, if you have identified a correct corner and a clockwise cycle of the other three is needed, instead of doing the corner cycle twice (three times returns the cube to where it was), save time by reversing the sequence:reverse corner cycle:(L- U) (R U- ) (L U) (R- U-)Step 6:Every iteration of the corner rotator exchanges UFR and DFR, and repeating it 6 times returns the cube to where it was. Step 6A will use the corner rotator 2 times if the top color is on the right side of the UFR corner, or 4 times if it is on the front, in which case it is faster to do the reverse sequence 2 times (easy, start withD-instead ofR-and everything follows):reverse corner rotator:D- R- D R

Step 6 is the same as Step 3 of the solution presented forRubik's 2x2x2, and we repeat here the observations from that page:

- Step 6A affects only 4 corners by exchanging two front right corners and also exchanging the two back down corners.

- Doing Step 6A twice leaves corners in the same positions, except those four corners are rotated, and doing Step 6A six times leaves the corners the same as when you started.

- On the up layer Step 6 only modifies the front right corner.

- Since Step 6 started with the down corners correct, once three of the four up corners have been fixed, fixing the fourth up corner must leave the down layer correct. This is because when at every 6th move the two back down corners are correct, all that is left that could be incorrect are the two front right corners, but due to parity considerations, a completely solved puzzle except for two adjacent corners is not possible (however, although not hard to overcome, this is not true for the
Rubik 3x3x3 Void Cube).

- The only edge pieces that are affected are FR, RD, BD, which are on the lower two layers; they return back to where they were after 6 moves.

It is interesting to see that the Corner Rotator can be used for Step 5, by memorizing two "do simple nothing" sequences:

5 (alternate).Position the up layer corners:Let S be the sequence of Step 6A, and let X be the sequenceAlthough this sequence is relatively long after expanding eachS S S- S-(which does nothing) interleaved with rotating the up layer 360 degrees withU U U:^{2}alternate corner cycle:Z = S U S U S- U^{2}S-Zdoes a counterclockwise cycle of UFR, UBR, UBL; repeat it until at least one up corner is correct (but may be rotated), re-position the cube so this corner is UFL, and then continue repeating it until all up corners are in their correct positions.Sto the corresponding four moves, it is relatively easy to remember as the interleaving of two do-nothing sequences. In addition, if you forget what it does, a pencil and paper can be used to draw what happens to the up layer; see the explanation and diagrams presented for theRubik's 2x2x2Rubik 2x2x2 alternate solution.

A completely solved 3x3x3 cube except for two adjacent corners exchanged is not possible due to parity considerations; that is, if just two adjacent corners are interchanged, then it must be that the edges are not completely solved. However, this is possible for theRubik's 2x2x2cube.

If you are fast with the Step 5 corner cycle sequence, and don't want to bother remembering the sequence to exchange two corners of a 2x2x2 cube (which corrupts edges when used with the 3x3x3 cube), Steps 1, 5, and 6 will solve the 2x2x2 cube, by using the corner cycle appropriately for Step 5:

The single swap is all that is needed, since it can be used 3 times for a diagonal swap and twice for a double swap. However, by using all three variations shown above, it is at most 9 moves for a single swap (8 moves for the corner cycle plus the finalU) or 16 moves for the diagonal or double swap. Note each time a corner sequence or reverse corner sequence is done, the cube first needs to be repositioned so that the corner that does not move is in the UFR position. If you don't want to remember the reverse corner cycle, the second sequence of the double swap can be a standard corner cycle on D,B,A followed by aU; however, counting that final^{2}Uas two moves, it is no fewer moves than doing two single swaps.^{2}

Here is a different approach that starts with solving the corners, then the top and bottom edges, and finally the middle edges.

1.Solve the corners using a solution forRubik's 2x2x2.

2.Position up and down edges by moving to and from the middle layer:

- Cycle edges between the middle and up layers to get three top edges correct:

RB -> FU, FU -> FD, FD -> RB:F M F-

That is, repeatedly position the cube so that the edge to be moved is RB, rotate the U layer so that where you want to move it to is FU, and cycle.

- Turn the cube over, and repeat Step A.

- Move the edge that goes to FD to the FU position; then move final edge to FU.
3.Use this to flip up and down edges (parentheses are just to make the sequence easier to read):Flip the UF edge:F- M (F M)^{2}F-4.Use rotations of the middle layer and these to position middle edges:Front back swap,LF <-> LB, RF <-> RB:(R^{2}M^{2})^{2}

Clockwise cycle,RF -> LB -> RB -> RF:(R^{2}M) (R^{2}M-)

Although not necessary, these can save time:Diagonals swap,LF <-> RB, RF <-> LB:M L^{2}R^{2}M- L^{2}R^{2}

Counter clockwise cycle,RF -> RB -> LB -> RF:(M- R^{2}) (M R^{2})5.Use this to flip middle edges (for right to left diagonal, do B2 before and after):Flip RF and RB:(R M-)^{3}R M^{2}R (M- R)^{3}

Although not necessary, this can save time:Flip RF and LB:(R M-)^{3}(R M) (R M-)^{3}(R M)

*** Don't bother with this until you are good at the standard 3-step solution; the same notation is used here.

It is interesting that thecorner cycleandcorner rotatoroperations from the standard 3-step layer by layer solution can be used for an edges first approach to solving. The sequencerotates the three edges around the upper right by doing:S = U- R U R-UF -> UR, UR -> FR, FR -> UFIt leaves the UF and UR edges in the same flip orientation, and flips the flips the FR edge when it moves up. It has the side effect of permuting the UFL, UFR, UBR corners, but that won't matter because here we will be solving corners last.

GivenS, here is an outline of an edges-first approach to solving:

1.Solve the top layer edges, and turn the cube over (easy with a little practice).

2.Solve the middle layer:Rotate the middle so centers are correct, and then useSto place each of the edges.Scan be used to move an edge up to the top to then go to a different middle position. Also, if an edge needs to be flipped before placing it, rotate the up layer and position the cube so you want to move UB to RB with a flip, and doSin reverse:T = R- U R U-3.Solve the top edges:Scan be used to permute edges; for example:If some edges need to be flipped, some clever playing around withS U- S U-does the cycle UF -> UL -> UB -> UFSandTshould work.4.Position the corners:First, use thecorner cycleoperation to get the correct corners on the top and bottom (by temporarily positioning a side face to be up), then use it on the up and down faces.5.Rotate the corners:Use thecorner rotatoroperation to finish the up and down faces.

In time, for smoother operation, cubes were sold with redesigned central axes and adjustable springs in the centers; here is an example:

Plastic, stickerless, made in China, purchased from Amazon.com in 2015.

(left:Newisland, sold byYaMiYo, comes in a with a storage bag, 3.25" square;

right:DaYan, sold byMaxin, comes in a fitted box, 2.3" square)

In the early 2000's, smoother working versions of Rubik's 3x3x3 were widely available, with screws / springs for adjustable tension and smooth turning even when layers are not exactly aligned (beveled interior corners in conjunction with the spring action give a minimal degree of automatic alignment). TheNewislandcube shown above was a gift from a friend who does speed cubing; it is smooth and quiet, comes with a storage bag and directions, and its literature explains PA plastic lower resistance, anti-popping, and internal construction. The less expensiveDa Yancube shown above has different but similar construction; here are photos of it apart:

Here are two examples of more expensive cubes advertised for speed cubing.

In addition to complex mechanisms that include screws and springs, the Valk cube shown on the left below has magnets to give a slight click stop effect. The GANs cube on the right below has a similar weight and smooth action as the Valk, but with no magnets.

Both are beautiful cubes that are pleasurable to use, even if you are not speed cubing.

"Valk 3", designed by Mats Valk,

purchased from Amazon.com 2017.

(plastic, 2+3/16" square;

3.75" square box with a magnetic lid

and extra stickers and springs)

"GANS 356 Air Advanced,

purchased from Amazon.com 2018.

(plastic, 2+3/16" square;

with display box and nut tool,

and 4 pages of instructions 1, 2, 3, 4)

The original Rubik cube as well as modern versions are all based on a center spindle assembly that connects the six center squares and holds the whole cube together where the other pieces flow around it.

TheRubik 3x3x3 Void Cubeis based on a completely different idea. There are no centers (one can pass their finger through the cube in all three directions).

TheRubik 3x3x3 Edges Only Cube(a.k.a. cornerless void cube) uses the same mechanism and eliminates the corner pieces as well.

One way to measure large is the dimension of the cube; for example cubes of size 33x33x33 have been made. But another measure is the physical size of the cube. This cube, made by Tony Fisher, is 1.57 meters (over 5 feet) tall:

McDonalds, 2.2"

Chex Cereal, 2.2"

Jack Daniels, 2.2"

UPS, 2.2"

Mickey Mouse, 2.2"

MatLab, 2.2"

Small Cube, 1.2"

Small Shiny Cube, 1.2"

Dice made by Volker (Germany), 2.2"

Assembly Cube, 2.2"

Beust's Page, from: http://beust.com/rubik

Bieber's Page, from: http://www.ronaldbieber.de/Fun/Rubik

Chess And Poker Page, from: http://www.chessandpoker.com/rubiks-cube-solution.html

Cheyer's Page, from: http://www.ai.sri.com/~cheyer/rubiks/rubiks.html

Dedmore's Page, from: http://www.helm.lu/cube/solutions/rubikscube

Dry Erase Board Page, from: http://www.thedryeraseboard.com

Fridrich's Page, from: http://ws2.binghamton.edu/fridrich/cube.html

Jaap's Page, from: http://www.jaapsch.net/puzzles/cube3.htm

Jasmine Page, from: http://peter.stillhq.com/jasmine/rubikscubesolution.html

Jeays' Page, from: http://jeays.net/rubiks.htm

JJuergen's Page, from: http://www.mathematische-basteleien.de

Marshall's Page, from: http://helm.lu/cube/MarshallPhilipp

McFarren's Page, from: http://www.geocities.com/abcmcfarren/math/rc/RubCub0.htm

Monroe's Page, from: http://www.alchemistmatt.com/cube/rubik.html

Nerd Paradise Page, from: http://www.nerdparadise.com/puzzles/333

Olefsky Puzzle Solver Page, from: http://www.puzzlesolver.com

Ortega and Jelinek Corners First Solution Page, from: http://rubikscube.info/ortega.php

Oxford ComLab Text Solution, from: ftp.comlab.ox.ac.uk

Petrus' Page, from: http://lar5.com/cube

Rob's Rubik Repair Page, from: http://www.roobik.com/cgi-bin/rubix/rubix.cgi

Rubiks.com Solution, from: http://www.rubiks.com

Scared Cat Page, from: http://www.scaredcat.demon.co.uk/rubikscube/the_solution.html

Shengshou Speed Cube Solution, from: http://www.speedsolving.com/wiki/index.php/Shengshou

Shon's Rubik's Place Page, from: http://www.rubiksplace.com

Still's Page, from: http://peter.stillhq.com/jasmine/rubikscubesolution.html

You Rubik Page, from: http://www.yourubik.com

WikiHow Page, from: http://www.wikihow.com/Solve-a-Rubik's-Cube-(Easy-Move-Notation)

Rubik Hungarian Patent, BE887875.

Rubik U.S. Patent, from: www.uspto.gov - patent no. 4,378,116

Sugden Patent, from: www.uspto.gov - patent no. 6,974,130

Sugden Design Patent, from: www.uspto.gov - patent no. D495,378

Scott Patent Application, from: www.uspto.gov - application no. 2010/0230897

God's Number is 20, from: http://www.cube20.org

Kociemba's Two Phase Algorithm and Cube Math, from: http://kociemba.org/cube.htm

22 Moves, from: http://www.springerlink.com/content/q088143tn805k124/fulltext.pdf

Speed Solving Page, from: http://www.speedsolving.com/wiki/index.php/Main_Page

Superflip, from: http://www.speedsolving.com/wiki/index.php/Superflip

Rubiks.com Page, from: http://www.rubiks.com

Rubiks.com Booklet, from: http://www.rubiks.com

Rubiks.com Diagrams, from: http://www.rubiks.com

Rubiks Cube Typesetting with TeX, from: http://www.ctan.org/pkg/rubik

Rubik's Home Cube Assembly Instructions, from: http://www.rubiks.com/World/Rubiks%20downloads.aspx

Philip K's Rubik's History Puzzle List, from: http://hjem.get2net.dk/philip-k/puzzles/puzzlist.htm

Cube Lovers Archive, from: http://www.math.rwth-aachen.de/~Martin.Schoenert/Cube-Lovers

Wikipedia Page, from: http://en.wikipedia.org/wiki/Rubik%27s_cube

Wikipedia - God's Algorithm, from: http://en.wikipedia.org/wiki/God%27s_algorithm

Wikipedia - solutions, from: http://en.wikipedia.org/wiki/Optimal_solutions_for_Rubik's_Cube