Solving NxNxN Cubes
I have been very interested in the Rubik's cube for quite a while. Even though I don't really understand how the number of combinations are determined or how some of the larger cubes work, it still fascinates. Doing a search on Google for rubik cube turns up millions of pages, so I am not the only one out there with a fondness for this puzzle. This page is a copy of a text file that I once found for solving the cube. It has been enhanced with a couple pictures, and still remains the simplest solution I can find that works for cubes from 2x2x2 through 7x7x7 (and further). Since it is so hard to find this on the web currently, I put a copy up on my web site. ## NxNxN Cubes in PerspectiveI have added spaces in the "Combinations" numbers so that they would wrap properly with your web browser...
Mechanism: 6 armed spider
Mechanism: 6 armed spider
Mechanism: grooved sphere (Sebesteny mechanism), 6 armed spider (Eastsheen mechanism)
Mechanism: 6 armed spider, unknown
Mechanism: Unknown
Mechanism: Unknown To put those last numbers into perspective, some scientists believe that there's
"only" about 10^79 protons, neutrons, and electrons in the entire ## Move NotationI'll be using 2 different notations. First is the standard notation used in just about every solution manual to date, which is F(ront), B(ack), L(eft), R(ight), U(p), and D(own), followed by a + (clockwise quarter turn), - (counter clockwise quarter turn), or a 2 (half turn). I'll further expand this with X, Y and Z for the centre slices. X slices are parallel to U and D, Y are parallel to L and R, and Z are parallel to F and B. Their rotations are as follows: X, Y, and Z can be ANY centre slice. Simply change it as appropriate to the pieces you're working on. The other notation was used in "Rubik's Revenge: The Simplest Solution" by
William H. Mason, which is in the format x D y. x and y are
vertical columns on the front face, starting at 1 on the left and N on the
right. D is the starting direction to turn the bottom face, which is Column 1 DOWN, bottom LEFT, coloumn 2 DOWN, bottom RIGHT, To summarise, the columns
To get back to where you were before, perform the above pattern twice more. If you're still uncomfortable with the pattern, try a few more:
Each sequence of moves in this notation represents the (x, y) coordinates of the piece we're focusing on, with (1, 1) in the bottom left corner. For those of you not familiar with coordinate geometry that this means is that the bottom left corner is (1, 1), the piece to its right is (2, 1), the piece just above it is (1, 2), the (corner) centre piece beside both of these is (2, 2), etc. Or, put more simply, for coordinates (x, y) means count off x pieces to the right and y pieces up, counting the bottom left piece as 1 in both cases. If this is confusing the heck out of you, here's a diagram to help you out, using the Professor's/Ultimate/Borg cube.
Along the way, I may refer to coordinates or a pattern with x or y instead of a number. Merely stick the needed number in place of the letter. For example, x L 1 could refer to 1 L 1, 2 L 1, 3 L 1, 4 L 1, or 5 L 1. Similary, 2 R y could refer to 2 R 1, 2 R 2, 2 R 3, 2 R 4, or 2 R 5. It just depends on what piece you need to move. N, as you've probably guessed, is merely the size of your cube, and N-1 is 1 less than the size of your cube. This heavily mathematical representation may seem to overcomplicate things, but trust me... you'll end up thanking me later. ## Solution PrefaceDuring this solution process, you will undoubtedly come across several sections
which simply don't apply to your cube, unless, of course, you've decided to
take the plunge and engage the Borg. Simply skip over those sections until you
come to something relevant. Here's a rundown of the various pieces and the
minimal cube size which uses them, as well as the name of the sections dealing
with them. Note that not all pieces are on all cubes above its minimal size.
All pieces with an odd minimum are
1 Corners. This should be easy enough to solve on your own by kludging at it for a while, but if not, I'll give you a few basic methods to kludge it together on your own. If you have middle centre pieces on your cube, it's easier to keep everything straight, but if not, don't worry. Just pick a corner to define as "correct" and take it from there. ## Top CornersIf the corner you need to fix is on the bottom side of the cube, then move the bottom slice so it's directly below where it should go. If possible, hold the cube so that the corner's top colour is on the front side. Then, if it's on the Left side: perform D+ L+ D- L-. However, if the top colour is on the bottom side, hold the cube so the proper cubelet is on the left or right side. Left side: perform L+ D- L- F- D2 F+. If the corner in question is in the right place, but oriented wrong, then hold the cube so that the corner's top colour is in the front. If it's on the: Left side: perform L+ D- L- D+ L+ D- L+. Right now, the top corners should have matching colours on each side. If not,
just pretend any errant pieces are oriented incorrectly and go back to 1.
Pocket Cube users, you're almost done! Skip down ( ## Top CentresIf a centre piece belonging in the top layer is on a side other than the top or the bottom, then hold the cube so it's on the front side. Now rotate the top side until there's an incorrectly coloured centre piece (henceforth referred to as a "gap") in the same relative position, as in the image below. Now perform X- Y- X+ Y+ OR X+ Y- X- Y+. If the needed centre piece is on the bottom, then position it so that if you could look straight through the cube, the centres would match up as in the image below. Perform Y- X- Y+ X- Z+ X+ Z- OR Y- X+ Y+ X+ Z- X- Z+.
## Top EdgesIf a top layer edge piece is on a Now, if the piece is on the Left: perform X- F+ X2 F- If, however, the edge piece in question is on a Then perform F+ X+ F- OR F- X- F+ Finally, if it's on the bottom edge and the top colour's on the bottom, eg. Then perform Y- D+ Y+ D- OR Y- D- Y+ D+ to change it so that now you can perform
step 3 as above. The top side is now complete, and if you've done it right, you should have a solid ring of colour around the top edges. If not, move any errant edge pieces out of their current position and move them to where they should be, using the above tips. Even numbered cube users, skip the next step and go onto the Middle Edges. ## Middle CentersIf your cube is and odd numbered cube and thus has a true centre on each side, then rotate the middle layer of the cube until the centres match the ring around the top layer. If you don't do this, you'll have plenty of headaches later when you come across "impossible" configurations and wonder how it could have possibly gone so horribly wrong. ## Middle EdgesFlip your cube around now so that the bottom side becomes the front side.
From here on in, this solution will become Find a piece you need to move and rotate the front side until its non-front side matches the colour of an adjacent side. Now rotate the cube until the side you want to move it to is on the bottom, eg. If the edge piece is on the: Left side: 1 R y See? I told you this was going to get less verbose! :-) ## Bottom CornersWelcome back, Pocket Cubers! For those of you who skipped over all the intervening stuff like I suggested, then rotate your cube until the bottom layer is now the front layer. Unless the 4 corners are in the proper positions already or will be simply by rotating the front face (if they can be, do so now), then if possible, rotate the front face until there is only one cubie in the correct position. Then, if the corners need to be rotated as: 1 R N N L 1 Those of you who get either 0 pieces or 2 pieces in the proper place no matter how you rotate the front layer, rotate it until you get 0 correct, and then hold the cube so the pieces need to be swapped as seen below: Then perform R+ D+ F+ D- F- R- OR L- D- F- D+ F+ L+. If you combine 1 R N and N L 1, rotating the cube to gain the correct alignment
before performing the second part, Pocket Cubers, ## Bottom EdgesOf course, they're not really the bottom now, but that's beside the point. OK, here we go... Rotate (1, x), (x, 1), (x, N) clockwise: x L 1. Eg. if you're using a Rubik's Revenge and x=2: Rotate (2, 1), (1, 2), (2, 4) clockwise: 2 L 1. The mirror image would be: Rotate (3, 1), (4, 2), (3, 4) counterclockwise: 3 R 4. To rotate the bottom and side centre edges, perform one of the patterns, flip the cube around so that the top is now the bottom, and then perform the other. For bottom and left centres, start with the first pattern. For bottom and right, start with the second pattern (the mirror).
Original Cubers, ## Corner CentresIf you encountered Rubik's ultimate revenge, then you know why this step was saved for last. If not, your time WILL come... trust me.
Rotate the front slice so that the necessary centre is over the a gap in the same relative position in the bottom layer, eg. Now, if the corner is in the: Top left: 2 R N-1. Don't feel that you're forced to maintain the same front side at all times. If
it's more convenient to use a different face as the front, then by all means,
reorient the front and move the cube around as much as you want!
Rubik's Revengers: ## Edge CentresThe same thing applies here. If another side would be more convenient, then
reorient and Then if it's at the: Top: 3 L 2 OR 3 R 4. Whew! It's been a long haul, but you've just become ## C O N G R A T U L A T I O N S !You've just joined the ranks of the elite, and completed the ultimate revenge! Give yourself a pat on the back. You deserve it! Heck, for completing these puzzles, you deserve a standing ovation! Before I go, here's a few patterns to try out. For everyone: (L+ R+ F+ B+)^3 For original cubers and above: (The "^2" means to do the part in parenthesis twice) L2 R2 U2 D2 F2 B2 Try making your own, completely original patterns, or combine the ones above. On the Rubik's cube or above, you can make all letters in the Roman alphabet. On the Ultimate puzzle, you're able to make all of the Japanese katakana. Try spelling your name on the cube, or perhaps spelling the name of your favourite anime. By picking up your cube, you've opened up a door to a wondrous paradise of patterns that you've never experience before. The only limit to what you can create is your imagination. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||