(Back to) Square One / Cube 21
 (Back to) Square One / Cube 21  Introduction: 1. Description 2. The number of positions Proof of positions counting formula 3. God's Algorithm Full God's Algorithm results 4. ProgramsSolution: 1. Getting into cube shape Diagram of shapes Method 1: Gathering corners Method 2 Method 3: Pairing the edges 2. Solving the piecesNotationA short note about odd permutations Method 1 Separate layers, then solve corners, then edges. Phase 1: Put pieces in their layers. Phase 2: Put the corners correct. Phase 3: Put the edges correct. Method 2 Solve corners, then edges. Phase 1: The corner pieces. Phase 2: Place the edge pieces in their correct layer. Phase 3: Place the edge pieces in each layer correctly. Phase 4: Fix the central layer. Method 3 Solve edges, then corners. Method 4 Separate layers, then solve each layer. Phase 1: Put pieces in their layers. Phase 2: Solve each layer separately. Method 5 Separate layers, solve parity, then each layer. Phase 1: Put pieces in their layers. Phase 2: Fix parities. Phase 3: Solve first layer. Phase 4: Solve second layer.Odds and EndsLinks to other useful pages:Courtney McFarren's Site has an excellent solution to the Square-1, as well as many other puzzles.Chris Eggermont's page. A good general text about the puzzle, with a list of shapes for getting it into the cube.Chris and Kori's page. This also has a set of pages with puzzle shapes.Matthew Monroe's page. He has illustrated Andrew Arensberger's partial solution (which was the first web solution but has disappeared now), making it much easier. It still does not deal with odd permutations however.The rec.puzzles archive has an awkward solution here.A Nerd Paradise has solutions for the various cubes, Pyraminx, Skewb and Square-1.Description:This puzzle is a cube consisting of three layers. The top and bottom layersare cut like a pie in 8 pieces; 4 edge pieces and 4 corner pieces, 30 and 60degrees wide respectively. The top and bottom layers can rotate. The middlelayer is cut in only two halves along one of the lines of the other layers.If there are no corner pieces in the way, you can twist half the cube 180degrees so that pieces from the top and bottom layers mingle.The puzzle is unique in that the two types of pieces intermingle. The edgeand corner pieces can freely move between the two outer layers. Of course,the puzzle will not necessarily be a cube shape if the pieces are mixed.The puzzle has six colours, each face has a single colour similar to theRubik's cube.Square-1 was patented by Karel Hrsel and Vojtech Kopsky on 16 March 1993, US 5,193,809.The number of positions:There are three categories of puzzle shapes. Both layers have 4 edges and 4 corners each. One layer has 3 corners, 6 edges, the other 5 corners 2 edges. One layer has 2 corners, 8 edges, the other 6 corners and no edges.There are 1, 3, 10, 10 and 5 layer shapes with 6, 5, 4, 3 and 2 corners.This means there are 5·1+10·3+10·10+3·10+1·5 = 170 shapes for the top andbottom layers. The middle layer has two shapes (half of it is assumed to befixed). This means that there seem to be 170·2·8!·8! = 552,738,816,000positions if we disregard rotations of the layers. Some layer shapes howeverhave symmetry, and these have been counted too many times this way.To take account of the symmetries we can simply count the number of layershapes differently. Instead of the numbers 1, 3, 10, 10, 5 we use thenumbers 2, 36, 105, 112, 54, which are the number of shapes if we considerrotations different (e.g. a square counts as 3 because it has three possibleorientations). By the same method as before we then get 19305·2·8!·8!or 62,768,369,664,000 positions. To exclude layer rotations, divide by 122to get a total of 435,891,456,000 distinct positions.It is interesting to note that this number is exactly 15!/3. In fact, evenfor the generalised version with 2n pieces in each layer, a similarly simpleformula holds, viz. 4(4n-1)!/3n. I had proved this in arather complicated unsatisfactory way, but Mike Godfrey found aneat insightful proof. He even generalised it to8(C+E-1)!/(2C+E), where C and E are the total number of corner andedge pieces, and E>0.If instead we wish to count only all those positions where there are nocorner pieces in the way of twisting the halves, then we can use the samemethod but counting only all the different ways each shape can be split intotwo halves, e.g. a square counts as 2 this time. The numbers to plug in arenow 1, 12, 46, 62, and 37 which gives a total of 3678·2·8!·8! =11,958,666,854,400 twistable positions.God's Algorithm:In 2005 Mike Masonjones calculated God's Algorithm for the Square-1, inthe twist metric. This shows that every position can be solved in atmost 13 twists (12 if the middle layer is ignored). Thefull results for each shape are on a separatepage, but the totals can be seen in the table below.TwistsPositions01164211533170504235144530914586388932307452031138844591675049336710647701014950231093611183662070768126394512003213157452752Total: 435891456000Programs:To help solve this puzzle I have written computer programs that searchfor solutions for a particular position. The first Square-1 Solveris similar in style to Kociemba's algorithm for the Rubik's cube, as itsolves it in two stages. In the first stage it makes a cube with an evenpermutation of pieces, and the second stage solves it. With some simplelook-up tables the search is speeded up. This program has helped meimprove some of the move sequences in my previous solution method, andhelped me find many others.If you have more than 75 Megabytes of Ram, then you are better off withmy second program, the Square-1 optimiser. It searches for optimalsolutions only, using a single phase search with larger pruning tables.This fast program helped me turn all the sequences on this page intooptimal ones.From the results of the two programs I suspect that God's algorithmuses at most about 30 moves in total. Mike Godfrey has optimally solvedover 300,000 positions, and found two positions at depth 30.Square-1 solver, Version 3, for DOS/Windows.Square-1 optimiser, Version 1, for DOS/Windows.Solution:Section 1: Getting the cube shape.First you must get the puzzle into cube shape. This takes at most 7 twists(ignoring the centre layer shape), and the shortest method is simply using atable to look up the moves since there are only 90 different possiblepuzzle shapes. Such tables can be found at various websites, but they are notparticularly enlightening. Instead I will present three methods here. The firstis the most intuitive, and is the most common way of doing this phase. The secondmethod is very different, but interesting as it shows an underlying mathematicalconstraint that I find quite surprising. The third is a little similar to the firstthough described in much more detail.I have made a diagram showing the variousshapes. At the top is the solved cube shape, and each successive row below it is onemove further away. The blue shapes have 6 corners in one layer and two in the other,and these shapes are used in method 1 below. The red shapes have five and threecorners in the layers. The yellow, green and purple ones have 4 corners in each layer,and the three colours signify the type of shape as used in method 2.Method 1: Gathering cornersThis method simply involves bringing six of the corners into one layer, which leavesthe remaining pieces (2 corners and 8 edges) in the other layer. From such a position itis fairly easy to recover the cube shape.I will not go into detail on bringing 6 corners together as it is quite easy. Onceyou have three corners together in half of one layer, then move these pieces around tothe left half of the bottom layer. Now keep them there, so only use twists of the righthand side and turns of the top layer, and try to bring three more corners together inthe top layer. Finally bring these two sets of three together in one layer.Hold the puzzle with the six corners in the bottom layer. The top layer has two cornersand six edges. There are only 5 possible shapes it can have, because there must be anywherefrom 0 to 4 edge pieces between the two corner pieces. The diagrams below show how the cubeshape can be recovered in each case. 0 /(-2,-4)/(-1,-2)/(-3,-3)/  1 /(2,-2)/(-3,-4)/(4,-3)/(-5,-4)/(6,-3)/  2 /(-4,-2)/(-1,4)/(-3,0)/  3 /(-4,0)/(5,4)/(2,-3)/(-5,-4)/(6,-3)/  4 /(2,2)/(0,-1)/(3,3)/  The notation used here will be explained later when it will be more extensivelyused instead of diagrams.[ Skip to Section 2: Solving the pieces ]Method 2:This is a method found in the rec.puzzles archive.There are 10 shapes with 4 edges (e), 4 corners (c) that a layer can have.Looking clockwise around the layer, they are:  cececece: Square0  ccecceee:Mushroom0  ccceecee:Shield0  cceecece:Right Fist 1  ccceceee:Left Paw1  cceeccee: Barrel2  cccceeee:Scallop 2  cceceece:Kite2  ccececee:Left Fist3  ccceeece:Right Paw 3If you are in a position with two of these shapes, look up the numbersassociated with them in the list above and add them to get a number.Whatever move you do that keeps 4 corners in each layer, the number arrivedat will not change (modulo 4). In particular if the number is not 0 (or 4)you cannot get it into a cube shape without first moving to a position thatdoes not have 4 corners on each side. You then have to move back to aposition with 4 corners in each layer and that does have a 0 or 4 sum.If you do get a 0 (or 4) sum then the following list shows you how far awayyou are from the cube shape: Square-Square 0 Kite-Kite 1 Barrel-Barrel 2 Shield-Shield 3 Scallop-Scallop 3 Mushroom-Mushroom3 Barrel-Scallop 4 L.Fist-R.Fist 2 L.Paw-R.Paw 3 Barrel-Kite 3 Square-Shield 3 Square-Mushroom3 Kite-Scallop 3 R.Fist-R.Paw 4 L.Fist-L.Paw 4 Mushroom-Shield4 In the diagram of all the various shapes,the types of position can be seen. The yellow shapes have sum 0, the green have sum 2,and the purple have sum 1 or 3. The mirror image of a shape with sum 1 will have sum 3and vice versa. Only one of each such pair is shown in the diagram. Note that shapeswith only one non-symmetric layer will always have sum 1 or 3.[ Skip to Section 2: Solving the pieces ]Method 3: Pairing the edgesI am indebted to Robert Richter for the following method of restoring thesquare 1 puzzle to cube form. I have worked out his method in great detail,because I find it an interesting approach. Once you develop a feeling for it,it becomes easy and you will not need to remember any move sequences.The main idea is to combine all the edge pieces into adjacent pairs of edges.Once this has been done, it is easy to bring all four pairs together and fromthere get the cube shape using the first move sequence in the first method above.The great advantage of having only pairs of edges is that they take exactly thesame amount of room as corner pieces. Each layer therefore will consist of sixparts of equal size (corners and/or edge pairs) which can be moved around withoutobstruction.I'll explain below in more detail how to make edge pairs, but before I doit is important to be able to distinguish between edge pairs and singleedges. This is usually really obvious, but sometimes when there are an odd numberof edges together in a group it is not immediately clear which of them is thesingle edge and which ones form the pairs. A simple way to determine this isto remember that the single edge is always one of those at the outside of thegroup, never one in the middle. Often one of those outside edges cannot bemoved because the seam that separates if from the others does not extend allthe way through the layer. As it cannot be separated, it must form a pair withthe one next to it, and the edge at the other end of the group is the singleone. If both edges can be moved then either one of them (but not both) can beconsidered to be single.Another way to determine the single edges is by seeing how close the layeris to being made of six equal sized parts. There are several shape types: The layer is square, and obviously there are 4 single edges in this case. The layer does divide into 6 equal sized parts. Here all edges (if any) already occur in pairs. The layer has 4 of the 6 equal parts, but the rest consists of two edges with a corner between them. Those two edges are the single ones. Half the layer consists of 3 equal parts, and the rest consists of two edges with two corners between them. Again those two edges are single. Sometimes alayer is ambiguous, because it belongs to both cases 3 and 4 above. In particular,the layer shape CCECEEEEE can be split as EE-EE-C-C-ECE, or asC-EE-EE-ECCE. We will use this ambiguity later on.So, the first thing to do to move towards the cube shape is to pair up theedge pieces. Getting three pairs is fairly easy, in fact, in more than halfthe configurations there are already three pairs. In nearly all other casesthere will be unpaired edges in both layers. It is then possible to bring anunpaired edge in the top layer to the front to the left of the seam, and anunpaired edge in the bottom layer to the back to the right of the seam so thata single twist brings the two edges together and does not split any alreadypaired edges. This process can be repeated if necessary until you have atleast three edge pairings. There is one special case you should be aware of,where all unpaired edges are in a single layer which only occurs when it isa square shape. In this case simply do any twist (that does not split anypairs in the other layer) to spread the unpaired edges to both layers andthen continue the process outlined above to bring them together.You will now have at least 3 pairs of edges. There are now threepossibilities left: You already have 4 pairs of edges. It is now a simple matter to bring these pairs together to form a large group of 8 adjacent edges.  You have two unpaired edges which are 150 degrees apart, i.e. they are separated by two corner pieces. Turn the layer with these two edges so that they both lie on the left hand side of the seam. I prefer to have them at the bottom left so I turn over the puzzle if they are on top. You next use only turns of the top layer and twists of the right hand side to move the three edge pairs together in a large group of 6 edges in the top layer. Often this is easy. The trickiest case is when the bottom right half consists of an edge pair between two corners. This pair cannot be immediately placed next to another edge pair, so you will have to twist it to the top and bring an edge pair to the bottom so that it lies next to the seam. Just be patient and make sure not to turn the bottom layer or to break up the edge pairs. Once you have the large group of 6 edges you can place them next to the two single edges to make a group of 8 edges together.  You have two single edges that are 90 degrees apart, i.e. they are separated by exactly one corner piece. Turn the layer with these two edges so that they both lie on the same side of the seam, and turn the puzzle so that they are in the left half of the bottom layer. Using only turns of the top layer and twists of the right half of the puzzle bring (at least) two of the edge pairs together in the top layer to make a group of 4 (or 6) adjacent edges. It is now always possible to bring the single edges and the group of 4 together in one layer to make the layer shape CCECEEEEE (or its mirror image CCEEEEECE), and this leaves (or moves) the remaining edge pair to the other layer. Because of the ambiguity of the edge pairings in this shape, you can consider the two edges on either side of the two adjacent corners to be the unpaired ones instead. This can therefore be solved as in case 2 above. [Note: You could use the ambiguity of the CEEECCEEE layer shape instead but this it is slower to solve because its edge pairs are separated, and you need to put all three edge pairs together in case 2.]At this point all eight edges are together in one layer. A single simple sequenceof moves now recovers the cube shape. Split the group of edges in two, making bothlayers into the scallop shape (CCCCEEEE). Now split each set of edges in two againto get both layers into the barrel shape (CCEECCEE). Split one barrel between a pairof corners and the other between a pair of edges, to get both layers into the kite shape(CECEECEC). If you split these between their edge pairs you form two squares. The above should give you enough information to eventually get two squares.If the middle layer is not square, a single twist can make it so, thoughit is not yet necessary to do this at this point. The middle layer could wellget disturbed in the later stages, but it can always be solved independentlyat the end.Section 2: Solving the pieces.Notation:From now on we need a notation for the moves. Hold the puzzle with the'square 1' text on the left, so that the cut in the middle layer goes fromthe front left to the back right. I hold my left thumb on the small front-leftface of the middle layer, and my left fingers on the back. I use my right handto turn the top or bottom layers, as well as the right hand half of the puzzle.Turns of the top and bottom layers are denoted by a pair of numbers (n,m).They are the multiple of 30 degrees that the top/bottom layers are to turnrespectively. A 3 means clockwise 90 degrees, 1 means 30 degrees (i.e. oneedge along) and negative numbers mean anti-clockwise turns. A twist of theright hand side of the puzzle is denoted by a /. As the puzzle is in the cubeshape now, the move sequences (1,0)/(-1,0) or (0,-1)/(0,-1) for example bothleave the puzzle in cube shape if you ignore the middle layer.As a shorthand, I will also use letters to denote the possible positionsof the pieces in the cube. This notation is based on that of the Rubik's Cube.Let the faces be denoted by L, R, F, B, U and D (left, right, front, back, up,down). Positions of edge pieces can be denoted by two letters, for example theedge position on the left of the top layer is denoted UL. Similarly positions ofcorners are denoted by 3 letters, for example DFR is the front-right cornerposition of the bottom layer.A short note about odd permutations:Now that the puzzle is a cube shape, it is tempting to think that it can nowbe solved without changing shape. Many of the sequences used below indeed keepthe top and bottom layers square. Each shape preserving move swaps two pairs ofcorners and two pairs of edges. This is an even number of swaps, so this is aneven permutation. No combinations of even permutations can possibly be an oddpermutation so any position that needs an odd number of swaps to solve it, cannotbe solved without changing shape. In particular, to make a single swap of piecesyou must change the shape of the puzzle. The quickest way to perform an odd permutationis to go to the scallop-scallop shape (3 twists), swap three corners from onelayer with three from the other (1 twist), and return to the cube shape (3 twists)which takes 7 twists in total.There are several methods for placing the pieces into their correct positions.I will show the most common ones here.Method 1:This first method was worked out and given to me by Bobfish. The pieces are firstput into their own layers, then the corners are placed, and finally the edges.Some of the sequences used in this solution can be made slightly shorter. If youwant, you can look them up in the computer generated tables that are listed inmethod 2 below. I have left this solution as is, because Bobfish found thesesequences all by hand without computer. The parity fixer in particular isespecially impressive - it is a much better clean odd edge swap than I ever foundby hand myself.[ Skip to Method 2: Corners first ]Phase 1: Put pieces in their layers.This phase puts all the white pieces in the top layer, and all the green pieces inthe bottom layer. You can always keep the top an bottom layers square throughoutthis phase. First put all the green corners in the bottom layer. This is quite easy if you first put two green ones in the left half of the bottom layer, then pair up the other two green corners and also twist them into the bottom layer. You can now make the middle layer square if necessary by doing the move sequence /(6,0)/(6,0)/(6,0). All move sequences later on will have an even number of twists, so will leave the middle layer square. Now bring the edges to their layer. There are 8 possible cases, which are listed below. In each case I will list the positions where there are edges in the wrong layer. 1. All edges are in the correct layer: do nothing. 2. UB, DL: do moves (1,0)/(-3,0)/(-1,-1)/(3,0)/(1,-2)/(-3,0)/(-1,-1)/(4,1)/(-1,0) 3. UF, UB, DF, DB: do moves (1,0)/(-1,-1)/(0,1). 4. UB, UL, DR, DB: do moves (1,0)/(-3,0)/(-1,-1)/(4,1)/(-1,0) 5. UB, UL, DF, DB: do moves (1,0)/(3,0)/(-1,-1)/(-3,0)/(4,1)/(-3,0)/(-1,-1)/(4,1)/(-1,0) 6. UF, UB, DB, DL: do moves (0,-1)/(-3,0)/(1,1)/(3,0)/(0,-3)/(-3,0)/(-1,-1)/(4,1)/(-1,0) 7. UF, UR, UB, DF, DB, DL: do moves (1,0)/(-1,-1)/(-2,1)/(-3,0)/(-1,-1)/(3,0)/(1,-2)/(-3,0)/(-1,-1)/(4,1)/(-1,0), or apply sequence 3 and then sequence 2. 8. All edges wrong: do moves (1,0)/(-1,-1)/(-2,-2)/(-1,-1)/(0,1), or apply sequence 3 twice.Phase 2: Put the corners correct.This phase puts all the corners correct, while leaving the all edges in their own layers. This phase can be done using only two move sequences: 1. Swap UFR-UBL, DFR-DBR: /(3,0)/(-3,0)/(3,0)/(3,0)/(6,0)/ 2. Swap UFR-UBR, DFL-DBR: /(0,-3)/(0,3)/(0,-3)/(0,-3)/(0,6)/ If you want to keep the layers square, you may do (1,0) before performing these, and (-1,0) at the end. The second sequence is actually the same as the first performed upside down. You will probably have to apply these sequences more than once before the corners are solved. If you only need a single corner swap, then give that layer a quarter turn. Its corners now need a three-cycle, which can be done by twice swapping adjacent corners.Phase 3: Put the edges correct. Most of this phase can be done using only two move sequences: 1. Swap UF-UB, DF-DB: (1,0)/(-1,-1)/(6,0)/(1,1)/(-1,0) 2. Swap UL-UF, DR-DF: (1,0)/(-3,0)/(-1,-1)/(3,0)/(1,1)/(-3,0)/(-1,-1)/(4,1)/(-1,0) You will probably have to apply these sequences more than once before the edges are mostly solved. These sequences all perform two swaps. It is possible that you need an odd number of swaps to solve the edges. To this end you will have to fix the parity by using the following move sequence: 3. Swap UF-UB: /(3,3)/(1,0)/(-2,-2)/(2,0)/(2,2)/(-1,0)/(-3,-3)/(-2,0)/(3,3)/(3,0)/(-1,-1)/(-3,0)/(1,1)/(-4,-3)Method 2:This method solves the corners completely, then the edges in their layers, and finally the edges completely.[ Skip to Method 3: Edges first ]Phase 1: The corner pieces.In this phase, every twist will keep the layers square. Find the piece that should be at UFL. If it is in the bottom layer, then bring it to the top with a twist, (keeping the U,D layers square by turning either layer slightly before twisting). Then turn the top to place it correctly. Find the piece that should be at UBL. If it is incorrectly placed in the top layer, then bring it to the bottom with a twist. Then turn the bottom to place it at DFR. Now do (4,0)/(-4,0) Find the piece that should be at UBR. If it is incorrectly placed in the top layer, then bring it to the bottom with a twist. Then turn the bottom to place it at DFR. Now do the twist (1,0)/(-1,0). Find the piece that should be at UFR. If it is not in place then turn the bottom to place it at DFR. Now do the sequence (1,0)/(0,-3)/(0,3)/(-1,0). Rotate the bottom layer to place at least 2 corners are correct. If you need to swap two corners, then you can use the following two sequences: 1. Swap DFR-DBR: (1,0)/(0,-3)/(3,0)/(-3,0)/(-3,3)/(-3,0)/(2,0) 2. Swap DFL-DBR: (1,0)/(3,3)/(0,3)/(3,3)/(0,3)/(3,3)/(-1,-3) It is possible to perform the second (diagonal) swap by using the first (adjacent) swap twice.Phase 2: Place the edge pieces in their correct layer.Rotate the top/bottom layers to bring it to a position where one of thefollowing sequences will place the edges in the correct layer. I have tried to includeas many useful cases as possible in this table, but there is no need to memoriseall of these. If you know only one of the sequences 8 to 13, you can use it repeatedly untilthe edges are in the correct layer, though using sequence 3 and 4 will speed things up. 1. Swap DF-UF, DR-UR, DB-UB, DL-UL (3,2)/(1,1)/(-3,3)/(-1,-1)/(0,-5) [4*|13*] 2. Swap DF-UF, DB-UB (0,5)/(1,1)/(-1,6) [2*|7*] 3. Swap DF-UB, DB-UF (0,-1)/(1,1)/(-1,0) [2*|6*] 4. Swap DR-UR, DB-UB (0,2)/(0,3)/(1,1)/(-1,-4)/(0,-2) [4*|11*] 5. Swap DR-UB, DB-UR (0,2)/(0,3)/(1,-5)/(-1,5)/(0,3)/(0,-2) /(0,3)/(0,-4)/(1,1)/(-1,3)/(0,-3)/ [5*|13*][6|13*] 6. Swap DB-UB, DR-UF /(-3,0)/(0,5)/(6,1)/(0,3)/(-5,0)/(-1,6) [6*|14*] 7. Swap DB-UF, DR-UB (1,0)/(0,5)/(0,-3)/(-1,6)/(0,-5)/(0,3)/(0,6) [6*|14*] 8. Swap DR-UF, UR-UB (1,0)/(0,-3)/(-1,2)/(1,1)/(0,-3)/(0,3)/(-1,0) [6*|15*] 9. Swap DR-UR, UF-UB (1,0)/(0,-3)/(-1,-4)/(1,1)/(0,3)/(0,3)/(-1,0) [6*|15*] 10. Swap DR-UB, UF-UR (0,-1)/(0,3)/(0,-3)/(1,1)/(-1,2)/(0,-3)/(0,1) [6*|15*] 11. Cycle UF->UR->DR (-5,-3)/(5,0)/(0,3)/(-5,0)/(5,0)/(0,-3)/(0,3) [6*|14*] 12. Cycle UF->UB->DR (1,0)/(-1,0)/(0,3)/(-5,0)/(5,0)/(0,-3)/(1,0)/(5,0) [7*|15*] 13. Swap UF-DF (0,-3)/(0,3)/(1,0)/(2,-2)/(4,0)/(4,0)/(-4,4)/(0,1)/(0,3)/(-3,-1)/(1,-2)/(-1,0) [11*|27*] Phase 3: Place the edge pieces in each layer correctly.Rotate the top/bottom layers to bring it to a position where one of thefollowing sequences will place the edges correctly. There is no need to memorise allof the sequences, as only a few of the single layer swaps will be sufficient. Alternativelyany odd permutation together with sequences a4 and a5 will do. The following sequences perform swaps in both layers: 1. Swap UF-UB, UR-UL, DF-DB, DR-DL (1,0)/(2,-4)/(1,1)/(2,2)/(-5,1)/(5,0) [5*|15*] 2. Swap UF-UL, UR-UB, DF-DL, DR-DB (0,2)/(-3,0)/(1,1)/(3,-3)/(-1,-1)/(0,3)/(0,-2) [6*|16*] 3. Swap UF-UB, UR-UL, DF-DL, DR-DB /(3,3)/(0,3)/(1,-5)/(4,-2)/(4,1)/(3,3)/ [7*|18*] 4. Swap UF-UB, DF-DB (1,0)/(5,-1)/(-5,1)/(5,0) [3*|9*] 5. Swap UR-UB, DR-DB (0,2)/(0,-3)/(1,1)/(-1,2)/(0,-2) [4*|11*] 6. Swap UF-UB, DR-DB (1,0)/(0,-1)/(0,-3)/(5,0)/(-5,0)/(0,3)/(0,1)/(5,0) [7*|15*] 7. Swap UF-UB, UL-UR, DF-DB /(3,3)/(0,-3)/(3,3)/(0,-2)/(-2,4)/(2,-4)/(0,-1)/(3,3)/(0,3) [9*|23*] The following sequences move top layer edges only: 1. Swap UF-UB, UR-UL /(3,-3)/(3,-3)/(0,1)/(-3,3)/(-3,3)/(-1,0) [6*|16*] 2. Swap UF-UL, UR-UB /(3,3)/(0,3)/(1,1)/(-1,-4)/(-3,-3)/ [6*|15*] 3. Cycle UF->UB->UR (1,0)/(-1,2)/(-5,1)/(0,3)/(-3,0)/(5,2)/(-5,4)/(-4,0) (-3,0)/(3,0)/(1,0)/(0,-3)/(-1,0)/(-3,0)/(1,0)/(0,3)/(2,0) [7*|19][8|17*] 4. Swap UF-UB (3,0)/(3,3)/(-1,0)/(2,-4)/(4,-2)/(0,-2)/(-4,2)/(1,-5)/(3,0)/(3,3)/ [10*|26*] 5. Swap UF-UR (0,3)/(-3,-3)/(0,-5)/(-2,2)/(0,4)/(0,-4)/(-1,2)/(3,0)/(3,4)/(0,-4)/(0,1)/(3,0)/ (0,3)/(-3,0)/(0,3)/(0,-3)/(0,3)/(2,0)/(0,2)/(-2,0)/(4,0)/(0,-2)/(0,2)/(-1,4)/(0,-3)/ [12*|28][13|27*] 6. Cycle UF->UL->UB->UR (1,0)/(2,2)/(0,-2)/(3,3)/(1,0)/(4,4)/(0,-2)/(2,2)/(0,-1)/(3,3)/ [10*|25*] 7. Cycle UB->UF->UR->UL (-3,2)/(1,-2)/(-4,0)/(0,3)/(1,0)/(3,-2)/(-4,0)/(-4,0)/(-2,2)/(-1,0)/(0,-3)/ [11*|26*] Phase 4: Fix the central layer. If the middle layer is not a square, then the following sequence will correct it by reversing its right hand side: /(6,0)/(6,0)/(6,0)Method 3: Edges First.First solve the edges completely, in the same way as phase 1 of method 2. Then put the corners in their layers andsolve them using the sequences in the tables below:[ Skip to Method 4: Layer method ] The following move corners from one layer to the other. 1. Swap UFR-DFR, UBR-DBR, UBL-DBL, UFL-DFL: (4,0)/(2,2)/(-3,3)/(-2,-2)/(-1,-3) [4*|13*] 2. Swap UFL-DFL, UBR-DBR: (4,0)/(2,2)/(6,-2) [2*|7*] 3. Swap UFL-DBR, UBR-DFL: (0,2)/(-2,-2)/(2,0) [2*|6*] 4. Swap UFL-DFL, UFR-DFR: (1,0)/(3,0)/(2,2)/(-5,-2)/(-1,0) [4*|11*] 5. Swap UFL-DFR, UFR-DFL: (1,0)/(3,0)/(-4,2)/(4,-2)/(3,0)/(-1,0) [5*|13*], or /(3,0)/(-5,0)/(2,2)/(3,-2)/(-3,0)/ [6|13*] 6. Swap UFL-DFL, UBR-DFR: /(3,0)/(0,-4)/(6,-2)/(0,-3)/(4,0)/(2,6) [6*|14*] 7. Swap UFL-DFR, UBR-DFL: (0,6)/(0,3)/(0,-4)/(4,0)/(-3,0)/(6,-2)/(2,0) [6*|14*] 8. Swap UFR-UBR, UFL-DFR: (0,2)/(0,-3)/(-2,1)/(2,2)/(0,-3)/(0,3)/(0,-2) [6*|15*] 9. Swap UFL-UBR, UFR-DFR: (0,2)/(0,-3)/(-2,-5)/(2,2)/(0,3)/(0,3)/(0,-2) [6*|15*] 10. Swap UFL-UFR, UBR-DFR: (-2,0)/(0,3)/(0,-3)/(2,2)/(-2,1)/(0,-3)/(2,0) [6*|15*] 11. Cycle UFL->UFR->DFR: (4,5)/(3,0)/(-4,0)/(0,4)/(0,-3)/(0,-4)/(0,-5) [6*|14*] 12. Cycle UFL->UBR->DFR: (-2,0)/(2,0)/(0,-3)/(4,0)/(-4,0)/(0,3)/(-2,0)/(-4,0) [7*|15*] 13. Swap UFR-DFR: /(3,3)/(-1,0)/(2,-2)/(0,-1)/(0,3)/(-2,-1)/(2,0)/(-1,-2)/(-3,0)/ [10*|23*] The following move corners within both layers. 1. Swap UFR-UBL, UFL-UBR, DFR-DBL, DFL-DBR: (1,0)/(-1,5)/(3,3)/(1,1)/(-3,3)/(5,0) [5*|15*] 2. Swap UFR-UFL, UBL-UBR, DFR-DFL, DBL-DBR: (1,0)/(0,-3)/(2,2)/(-3,3)/(-2,-2)/(3,0)/(-1,0) [6*|16*] 3. Swap UFL-UBR, DFL-DBR: (0,5)/(-2,4)/(2,-4)/(0,1) [3*|9*] 4. Swap UFL-UFR, DFL-DFR: (1,0)/(-3,0)/(2,2)/(1,-2)/(-1,0) [4*|11*] 5. Swap UFL-UBR, DFL-DFR: (4,0)/(0,2)/(0,3)/(-4,0)/(4,0)/(0,-3)/(0,-2)/(2,0) [7*|15*] 6. Swap UFL-UBR, UFR-UBL, DFL-DBR: /(3,3)/(1,2)/(2,-4)/(4,-2)/(-2,-1)/(3,3)/ [7*|19*] The following move corners in the top layer only. 1. Swap UFL-UBR, UFR-UBL: /(3,-3)/(3,-3)/(0,-2)/(-3,3)/(-3,3)/(2,0) [6*|16*] 2. Swap UFL-UFR, UBR-UBL: /(3,3)/(-5,-2)/(2,2)/(3,0)/(-3,-3)/ [6*|15*] 3. Cycle UFL->UBR->UFR: (1,0)/(0,3)/(2,0)/(-3,0)/(-2,0)/(0,-3)/(2,0)/(3,0)/(-3,0) [8*|17*] 4. Swap UFL-UBR: /(3,3)/(3,0)/(-4,2)/(-2,4)/(6,2)/(-4,2)/(-2,4)/(-5,0)/(-3,-3)/ [10*|26], or /(-3,0)/(0,-3)/(2,0)/(0,2)/(-2,0)/(4,0)/(0,-2)/(0,2)/(-4,1)/(3,0)/ [11|22*] 5. Swap UFL-UFR: /(3,3)/(1,4)/(0,4)/(-2,4)/(-4,2)/(2,0)/(-2,4)/(4,1)/(3,3)/ [10*|26], or /(0,3)/(1,2)/(0,-4)/(2,-2)/(0,-4)/(0,-4)/(0,4)/(0,-4)/(-3,0)/(0,-3)/ [11|23*] 6. Cycle UFL->UBL->UBR->UFR: /(3,3)/(-1,0)/(-2,-2)/(2,0)/(2,2)/(0,-2)/(2,2)/(-1,0)/(-3,-3)/ [10*|24*]Method 4: Layer method[ Skip to Method 5: Alternative Layer method ]Phase 1: Put pieces in their layers.This phase puts all the white pieces in the top layer, and all the green pieces inthe bottom layer. You can always keep the top an bottom layers square throughoutthis phase. First put all the green corners in the bottom layer. This is quite easy if you first put two green ones in the left half of the bottom layer, then pair up the other two green corners and also twist them into the bottom layer. Now bring the edges to their layer. There are 8 possible cases, which are listed below. In each case I will list the positions where there are edges in the wrong layer. 1. All edges are in the correct layer: do nothing. 2. UB, DF: do moves (0,-1)/(-3,0)/(4,1)/(-4,-1)/(3,0)/(0,1) [5*|13], or (-2,0)/(5,0)/(0,3)/(-5,0)/(5,0)/(0,-3)/ [6|12*] 3. UF, UB, DF, DB: do moves (-1,0)/(1,1)/(0,-1) [2*|4*] 4. UB, UR, DL, DB: do moves (1,0)/(0,3)/(-1,-1)/(0,-3)/(0,1) [4*|10*] 5. UF, UR, DF, DB: do moves (1,0)/(0,-3)/(0,3)/(2,-1)/(-3,0)/(0,1) [5*|12*] 6. UF, UB, DF, DR: do moves (0,-1)/(3,0)/(-3,0)/(1,-2)/(0,3)/(-1,0) [5*|12*] 7. UR, UB, UL, DF, DR, DB: do moves (0,-1)/(1,4)/(-1,-4)/(-3,0)/(4,1)/(-1,0) [5*|14*] 8. All edges wrong: do moves (-1,0)/(1,1)/(2,2)/(1,1)/(0,-1) [4*|12*], or apply sequence 3 twice.Phase 2: Solve each layer separately.The following table shows all sequences that solve the top layer for each possible case, except forthe ones already listed before, such as those in method 2, part 3b. The bottom layer can be solvedin the same way using the mirror images of these sequences (or by turning over the puzzle first). The following swap the diagonally opposite corners UFL and UBR, as well as the U layer edges as shown: 1. B-L: (4,0)/(3,0)/(3,3)/(3,0)/(5,0)/(0,3)/(3,3)/(0,3)/ [8*|18*] 2. B-R: (1,0)/(3,0)/(3,3)/(3,0)/(5,0)/(0,3)/(3,3)/(0,3)/(3,0) [8*|19*] 3. L-R: (-3,0)/(3,3)/(3,0)/(3,3)/(3,0)/(3,3)/ [6*|15*] 4. F-B: (1,0)/(3,3)/(3,0)/(3,3)/(3,0)/(3,3)/(2,0) [6*|16*] 5. R->B->F: (-2,-1)/(0,3)/(0,1)/(-4,0)/(4,0)/(4,0)/(0,4)/(-4,0)/(2,1)/(3,-3)/(3,0)/(-1,-2) [11*|27*] 6. R->L->F: /(3,-3)/(0,-3)/(-1,4)/(-4,0)/(0,4)/(4,0)/(4,0)/(-4,0)/(0,1)/(0,3)/(0,6) [11*|24*] 7. B-L, F-R: /(3,3)/(-1,4)/(4,-2)/(2,-4)/(0,2)/(4,-2)/(-5,1)/(3,0)/(3,3)/ [10*|26*] 8. L-F, B-R: /(3,3)/(0,3)/(5,-1)/(-4,2)/(-2,0)/(-2,4)/(-4,2)/(2,1)/(3,3)/ [10*|26*] The following swap the adjacent corners UBL and UBR, as well as the U layer edges as shown: 9. R-F: (1,0)/(3,3)/(-4,-1)/(1,4)/(0,-3)/(0,-3)/(3,0)/(0,-3)/(-4,0) [8*|20*] 10. B-R: /(-3,3)/(3,0)/(0,-3)/(0,3)/(0,-3)/(3,0) [6*|13*] 11. L-B: (1,0)/(-3,3)/(3,0)/(0,-3)/(0,3)/(0,-3)/(2,0) [6*|14*] 12. F-L: /(-3,-3)/(4,1)/(-4,-1)/(0,3)/(3,0)/(-3,0)/(3,0)/(3,0) [8*|19*] 13. L-R: (0,-1)/(1,1)/(3,0)/(-3,0)/(-1,-4)/(3,0)/(-3,0)/(-2,4)/(3,0)/(2,0) [9*|22*] 14. F-B: (3,0)/(3,0)/(3,3)/(3,0)/(4,0)/(0,3)/(3,3)/(0,3)/(2,0) [8*|19*] 15. R->L->F: /(0,-3)/(-2,0)/(-3,-1)/(-1,2)/(0,-4)/(0,4)/(1,-4)/(0,3)/(0,-5)/(0,3)/(3,0)/(5,0) [12|27*] or /(0,3)/(0,-3)/(-5,0)/(0,3)/(0,5)/(0,-4)/(4,0)/(-2,-2)/(0,2)/(0,1)/(3,0)/(0,-1)/(0,4) [13|27*] or (3,3)/(1,6)/(-2,4)/(0,2)/(0,5)/(0,3)/(-2,-1)/(4,0)/(-4,0)/(3,-2)/(-3,0)/(0,6) [11*|29] 16. B->F->R: /(3,0)/(1,2)/(2,0)/(-1,-2)/(0,3)/(-1,0)/(2,0)/(-2,0)/(-1,0)/(-3,-3)/ [11*|24*] 17. L->R->B: /(3,0)/(0,2)/(0,-3)/(3,0)/(2,0)/(0,4)/(0,4)/(-4,0)/(-1,0)/(-3,0)/(-3,-4)/(0,4) [12|25*] or (1,-1)/(0,3)/(2,1)/(0,4)/(4,-4)/(-2,3)/(0,3)/(-2,3)/(0,-4)/(-2,0)/(-3,0)/(5,1) [11*|29] 18. F->B->L: /(3,0)/(1,0)/(-2,0)/(0,5)/(0,-1)/(4,0)/(0,2)/(-4,1)/(0,1)/(0,5)/(-3,0)/(-1,1) [12|26*] or (4,-1)/(0,3)/(-3,0)/(1,-1)/(4,0)/(2,-2)/(4,0)/(4,0)/(-4,0)/(1,0)/(0,3)/(5,-2) [11*|27] 19. F-L, B-R: /(3,3)/(0,3)/(1,-1)/(-2,0)/(0,2)/(0,-4)/(2,0)/(-2,0)/(1,0)/(-3,-3)/ [11|24*] or (-3,-3)/(-3,-3)/(-4,5)/(-4,0)/(4,-2)/(-4,2)/(-2,0)/(5,1)/(-3,0)/(3,3)/ [10*|27] 20. F-R, B-L: /(3,3)/(-1,2)/(-4,2)/(4,0)/(-2,4)/(-4,2)/(-1,1)/(3,0)/(3,3)/ [10*|26*] 21. L-R, F-B: /(3,3)/(-1,0)/(2,0)/(0,4)/(0,-4)/(2,2)/(0,-4)/(2,0)/(0,1)/(3,3)/(6,0) [11|25*] or /(3,3)/(1,-2)/(0,4)/(-2,4)/(2,-4)/(-1,1)/(5,-1)/(-2,1)/(3,3)/ [10*|27] 22. F->R->L->B: (-3,0)/(-3,0)/(3,3)/(3,0)/(1,-2)/(-1,2)/(-3,-3)/ [7*|18*] 23. F->L->R->B: (4,0)/(-3,0)/(3,3)/(3,0)/(2,-1)/(-2,1)/(-3,-3)/(-1,0) [7*|19] or /(-3,3)/(3,-3)/(1,0)/(0,3)/(0,-3)/(0,3)/(0,-3)/(2,0) [8|18*]Method 5: Alternative layer methodPhase 1: Put pieces in their layers.This phase is the same as in Method 4.Phase 2: Adjust the permutation parities of the layers.This phase make sure that each layer is in itself an even permutation. This cuts down the number ofpossibilities, leaving fewer (and generally shorter) sequences in the last phases. It is however slightlytricky to recognize the parities quickly. Depending on the parities of the layer permutations, do one of the following: 1. Both layers even parity: Do nothing. 2. Top layer odd, bottom even: /(-3,-3)/(0,-1)/(2,-4)/(4,-2)/(1,0)/(-3,-3)/ 3. Top layer even, bottom odd: /(3,3)/(1,0)/(4,-2)/(2,-4)/(0,-1)/(3,3)/ 4. Both layers odd parity: (1,0)/(5,-1)/(-5,1)/(-1,0) Phase 3: Solve the first layer.The following table shows the sequences that solve the top layer for each possible case. Notethat these sequences may mix up the (unsolved) bottom layer. The following sequences move only corners: 1. Cycle UFL->UBR->UFR, (-2,0)/(0,-3)/(2,2)/(0,-3)/(-2,4)/(2,0) [5|13*] 1. Cycle UFL->UFR->UBR, (-2,0)/(2,-4)/(0,3)/(-2,-2)/(0,3)/(2,0) [5|13*] 1. Swap UFL-UBL, UFR-UBR /(3,3)/(1,-2)/(2,2)/(-3,0)/(-3,-3)/ [6|15*] The following sequences move only edges: 1. Swap UL-UF, UR-UB /(3,3)/(0,3)/(1,1)/(-1,-4)/(-3,-3)/ [6|15*] 2. Swap UL-UR, UF-UB (1,0)/(3,-3)/(2,2)/(3,3)/(4,-2)/(-1,0) [5|15*] 3. Cycle UF->UB->UR (0,-1)/(0,-3)/(1,1)/(0,-3)/(-1,5)/(0,1) [5|13*] 4. Cycle UF->UR->UB (0,-1)/(1,-5)/(0,3)/(-1,-1)/(0,3)/(0,1) [5|13*] The following sequences swaps the UFL,UBL corners and moves edges: 1. Swap UFL-UBL, UL-UF (-2,0)/(0,3)/(-3,-3)/(3,0)/(2,0) [4|10*] 2. Swap UFL-UBL, UL-UB (3,0)/(3,0)/(3,3)/(3,0)/(3,0) [4|10*] 3. Swap UFL-UBL, UL-UR (-2,0)/(-3,0)/(2,2)/(0,3)/(1,-5)/(2,0) [5|13*] 4. Swap UFL-UBL, UB-UR (1,0)/(3,3)/(2,-1)/(-2,1)/(-3,-3)/(-1,0) [5|15*] 5. Swap UFL-UBL, UR-UF /(3,3)/(1,-2)/(-4,5)/(3,3)/ [5|13*] 6. Swap UFL-UBL, UB-UF (1,0)/(2,-1)/(1,-2)/(2,-1)/(-5,4)/(5,0) [5|15*] 7. Swap UFL-UBL, Cycle UL->UB->UF->UR (-2,0)/(0,-3)/(3,3)/(2,-1)/(3,3)/(4,-2)/(-1,0) [6|17*] 8. Swap UFL-UBL, Cycle UB->UR->UL->UF /(3,3)/(1,4)/(2,-4)/(3,0)/(-3,-3)/ [6|15*] 9. Swap UFL-UBL, Cycle UR->UF->UB->UL (-2,0)/(3,3)/(3,0)/(5,-1)/(1,-2)/(3,3)/(2,0) [6|17*] 10. Swap UFL-UBL, Cycle UF->UL->UR->UB /(3,3)/(0,3)/(1,-5)/(-4,-1)/(3,3)/ [6|15*] The following sequences swaps the UFL,UBR corners and moves edges: 1. Swap UFL-UBR, UR-UF (-2,0)/(2,-1)/(4,-5)/(-1,2)/(4,-5)/(2,0) [5|15*] 2. Swap UFL-UBR, UB-UF (1,0)/(-3,3)/(3,-3)/(-1,0) [3|9*] 3. Swap UFL-UBR, UL-UR /(-3,3)/(3,-3)/ [3|7*] 4. Swap UFL-UBR, UB-UR (-2,0)/(2,-4)/(0,3)/(1,1)/(-3,0)/(2,0) [5|13*] Phase 4: Solve the second layer.Turn the puzzle over, and solve the top layer using the sequences in the table below. Thistable is very similar to the one in the previous phase, except that the move sequences aregenerally a bit longer, because these have to keep the bottom layer intact. The following sequences move only corners: 1. Cycle UFL->UBR->UFR, (1,0)/(0,3)/(2,0)/(-3,0)/(-2,0)/(0,-3)/(2,0)/(3,0)/(-3,0) (1,0)/(-1,2)/(-5,-2)/(-3,0)/(3,0)/(-4,2)/(1,-2)/(-4,0) [8|17*] [7*|19] 2. Cycle UFL->UFR->UBR, (3,0)/(-3,0)/(-2,0)/(0,3)/(2,0)/(3,0)/(-2,0)/(0,-3)/(-1,0) (-2,0)/(2,-1)/(-2,4)/(-3,0)/(3,0)/(-1,-4)/(-2,1)/(5,0) [8|17*] [7*|19] 3. Swap UFL-UBL, UFR-UBR /(3,3)/(1,-2)/(2,2)/(-3,0)/(-3,-3)/ [6*|15*] The following sequences move only edges: 1. Swap UL-UF, UR-UB /(3,3)/(0,3)/(1,1)/(-1,-4)/(-3,-3)/ [6*|15*] 2. Swap UL-UR, UF-UB /(3,-3)/(3,-3)/(0,1)/(-3,3)/(-3,3)/(-1,0) [6*|16*] 4. Cycle UF->UR->UB (-2,0)/(0,-3)/(-1,0)/(3,0)/(1,0)/(0,3)/(-1,0)/(-3,0)/(3,0) (-2,0)/(2,-1)/(4,1)/(3,0)/(-3,0)/(5,-1)/(-2,1)/(5,0) [8|17*] [7*|19] 3. Cycle UF->UB->UR (-3,0)/(3,0)/(1,0)/(0,-3)/(-1,0)/(-3,0)/(1,0)/(0,3)/(2,0) (1,0)/(-1,2)/(1,-5)/(3,0)/(-3,0)/(2,5)/(1,-2)/(-4,0) [8|17*] [7*|19] The following sequences swaps the UFL,UBL corners and moves edges: 1. Swap UFL-UBL, UL-UF (1,0)/(0,-3)/(0,3)/(0,-3)/(3,0)/(-3,3)/(2,0) [6*|14*] 2. Swap UFL-UBL, UL-UB (3,0)/(-3,3)/(3,0)/(0,-3)/(0,3)/(0,-3)/ [6*|13*] 3. Swap UFL-UBL, UL-UR /(0,3)/(3,3)/(0,3)/(4,0)/(3,0)/(3,3)/(3,0)/(5,0) [8*|18*] 4. Swap UFL-UBL, UB-UR (1,0)/(3,0)/(-3,0)/(3,0)/(0,3)/(2,5)/(1,4)/(3,3)/(2,0) [8*|20*] 5. Swap UFL-UBL, UR-UF /(3,3)/(1,-2)/(-4,-1)/(0,3)/(3,0)/(-3,0)/(3,0)/(0,-3) [8*|19*] 6. Swap UFL-UBL, UB-UF (-2,0)/(-1,-1)/(-3,0)/(3,0)/(1,4)/(-3,0)/(3,0)/(2,-4)/(-3,0)/(0,1) [9*|22*] 7. Swap UFL-UBL, Cycle UL->UB->UF->UR (1,0)/(0,3)/(0,-3)/(0,3)/(0,-3)/(2,0)/(-3,3)/(3,-3)/ (-2,0)/(3,3)/(2,-1)/(1,-2)/(0,3)/(-3,-3)/(-3,0)/(-1,0) [8|18*] [7*|19] 8. Swap UFL-UBL, Cycle UB->UR->UL->UF /(3,3)/(1,4)/(2,-1)/(3,0)/(-3,-3)/(0,-3)/(0,-3) [7*|18*] 9. Swap UFL-UBL, Cycle UR->UF->UB->UL /(-3,3)/(3,-3)/(-2,0)/(0,3)/(0,-3)/(0,3)/(0,-3)/(-1,0) (1,0)/(3,0)/(3,3)/(0,-3)/(-1,2)/(-2,1)/(-3,-3)/(2,0) [8|18*] [7*|19] 10. Swap UFL-UBL, Cycle UF->UL->UR->UB /(-3,0)/(3,3)/(3,0)/(1,-2)/(-1,2)/(-3,-3)/(-3,0) [7*|18*] The following sequences swaps the UFL,UBR corners and moves edges: 1. Swap UFL-UBR, UR-UF (1,0)/(2,2)/(3,0)/(0,-3)/(4,1)/(3,0)/(-3,0)/(5,-1)/(0,3)/(0,-2) [9*|22*] 2. Swap UFL-UBR, UB-UF (1,0)/(3,3)/(3,0)/(3,3)/(3,0)/(3,3)/(2,0) [6*|16*] 3. Swap UFL-UBR, UL-UR /(3,3)/(3,0)/(3,3)/(3,0)/(3,3)/(3,0) [6*|15*] 4. Swap UFL-UBR, UB-UR (1,0)/(3,0)/(3,3)/(3,0)/(5,0)/(0,3)/(3,3)/(0,3)/(3,0) [8*|19*] If you find that you have erred, and have a last layer with an oddpermutation parity, the following move sequence will fix it while keeping the bottom layer intact./(-3,0)/(0,-3)/(2,0)/(0,2)/(-2,0)/(4,0)/(0,-2)/(0,2)/(-4,1)/(3,0)/ [11|22*]Odds and Ends: To swap top and bottom layers, do /(6,6)/(-1,1) [4*|6*] To change the middle layer shape, do /(0,6)/(0,6)/(0,6) [3*|6*] Home Links Guestbook geovisit();
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