Extended Chart of Methods for Solving Rubik's

[1] CFOP/Fridrich [1] Petrus CFCE [1] Singmaster [1 p. 61] Roux [1] Zborowski [1] Tripod [1] [1] Zborowski–Bruchem Spiegel [1] Waterman [1] [1] EG (2×2 method)2 method) [1][2] Ortega Guimond (2×2 2method) method) [1] [1] Thistlethwaite [1] Kociemba

Scrambled = ⟨R,L,F,B,U,D⟩ R,L,F,B,U,D = R,L,F,B,U =⟨R,L,F,B,U⟩⟩⟨R,L,F,B,U,D⟩ [1] 0% solved – 43;252,003;274,489;856,000 positions

2×1×1 2 2cubegroup×1×1 block [1] Solves 27.5% Four corners Cross Solves 20.6% Four corners oriented † EOLine 2×1×1 2 2 block = R,F,U cubegroup×1×1⟨R,L,F,B,U,D⟩=⟨R,L,F,B,U⟩⟩ [1 p. 26][2] Solves 26.9% 13.6% solved – 91,538;631,268;761,600 positions All corners oriented Thistlethwaite I Solves 27.7% 27.5% solved – 170;659,735;142,400 positions 17.0% solved – 19,776;864,780;288,000 positions Solves 16.9% 1×1×1 2 3 block cubegroup×1×1† [1] 31.2% solved – 32;506,616;217,600 positions 2×1×1 2 3cubegroup×1×1 block [1] Four corners † Orient remaining four corners ⟨R,L,F,B,U,D⟩ =⟨R,L,F,B,U⟩commutator⟩ =⟨R,L,F,B,U⟩ = doubly even permutation [1 p. 27] Solves 19.3% Cross [1] 20.6% solved – 3,814;109,636;198,400 positions Solves 7.3% – 7 cases 1.5% solved – 21;626,001;637,244;928,000 positions 26.9% solved – 227;546,313;523,200 positions Kociemba I 2×1×1 2 cubegroup×1×13 block [1] EG [1] Exchange Solves 47.6% 46.8% solved – 28,217;548,800 positions Cross + one pair [1] One layer [1] Solves 21.3% – 128 cases C*LL OLL 2×1×1 2cubegroup [1] Solves 3.9% – 4 cases ⟨R,L,F,B,U,D⟩ =⟨R,L,F,B,U⟩R,L,F,B,U2,D2⟩ =⟨R,L,F,B,U⟩ 40.0% solved – 592,568;524,800 positions 53.0% solved – 1,672;151,040 positions Solves 14.3% – 42 cases Solves 7.3% – 7 cases 16.9% solved – 21,119;142,223;872,000 positions

Line + edges oriented = EOLine = ⟨R,L,F,B,U,D⟩ R,L,F =⟨R,L,F,B,U⟩⟩ [1 p. 26][2] Orient edges [1] All corners oriented in their U/D layer † 27.7% solved – 159;993,501;696,000 positions Solves 9.2% Cross + two adjacent pairs [1] Cross + two opposite pairs Four middle slices 20.9% solved – 3,390;319,676;620,800 positions 52.6% solved – 2,015;539,200 positions 52.6% solved – 2,015;539,200 positions Solves 22.6% Thistlethwaite II Two 1×1×1 2 3 blocks cubegroup×1×1† Solves 30.7% 61.9% solved – 29;859,840 positions 2×1×1 2 cubegroup×1×13 block + edges oriented Cross + three pairs [1] Four corners + other corners oriented † 56.0% solved – 440;899,200 positions 64.5% solved – 9;331,200 positions 27.9% solved – 141;263,319;859,200 positions

MGLS F2L = U group [1] PBL [1] Solves 23.0% 75.6% solved – 62,208 positions PLL 2×1×1 2cubegroup [1] Solves 14.1% – 5 cases ZZ F2L 2×1×1 3 3cubegroup×1×1 block [1] Solves 7.0% – 2 cases Solves 52.5% Solves 24.2% EOLS [1] J-EOLL [1] J-ELL = LLEF [1] J-PLL ⟨R,L,F,B,U,D⟩ =⟨R,L,F,B,U⟩R,L,F2,B2,U2,D2⟩ =⟨R,L,F,B,U⟩ CMLL [1] Solves 15.7% – 302 cases Solves 4.60% – 3 cases Solves 11.6% – 15 cases Solves 12.5% – 21 cases 47.6% solved – 19,508;428,800 positions Solves 14.3% – 42 cases Tripod ⟨R,L,F,B,U,D⟩ =⟨R,L,F,B,U⟩R,F⟩ =⟨R,L,F,B,U⟩ [1 p. 55] J-COLL 80.2% solved – 7,776 positions 59.9% solved – 73;483,200 positions Solves 7.29% – 7 cases J-CLL = CLL [1] J-EPLL Corners † = quotient of 2 ×1×1 2 2cubegroup×1×1 cube group Edges placed = 2×1×1 2 2cubegroup×1×1 cube group + even rotations CPEOLL [1] Solves 14.3% – 42 cases Solves 7.0% – 2 cases 35.0% solved – 5;885,971;660,800 positions 61.1% solved – 44;089,920 positions Solves 11.6% – 15 cases ⟨R,L,F,B,U,D⟩ =⟨R,L,F,B,U⟩M,E,S,Ra,Fa,Ua⟩ =⟨R,L,F,B,U⟩ [1 p. 36] F2L + edges oriented [1] F2L + corners oriented F2L + edges placed 63.3% solved – 15;925,248 positions 80.2% solved – 7,776 positions 82.9% solved –2,304 positions 82.6% solved – 2,592 positions One layer + corners Thistlethwaite III Two 1×1×1 2 3 blocks + four corners cubegroup×1×1† PEP-EOLL 67.3% solved – 2;580,480 positions Solves 22.8% 76.3% solved – 46,080 positions ⟨R,L,F,B,U,D⟩ =⟨R,L,F,B,U⟩R,F2⟩ =⟨R,L,F,B,U⟩ [1 p. 57] PEO-COLL = OCLL [1] PCO-EOLL = OELL [1] Solves 4.6% – 3 cases 78.8% solved – 14,400 positions Solves 7.29% – 7 cases Solves 4.60% – 3 cases J-OLL = OLL [1] Solve 11.9% – 57 cases PCO-CPLL PEO-EPLL ⟨R,L,F,B,U,D⟩ =⟨R,L,F,B,U⟩M,E,S⟩ = slice group =⟨R,L,F,B,U⟩† [1 p. 10] Solves 7.0% – 2 cases Solves 7.0% – 2 cases 85.3% solved – 768 positions

Tripod last step [1] F2L + oriented [1] ⟨R,L,F,B,U,D⟩ =⟨R,L,F,B,U⟩R2,L2,F2,B2,U2,D2⟩ = R2,L2,F2,B2,U2= ⟨R,L,F,B,U⟩⟨R,L,F,B,U,D⟩=⟩ = square group [1 p. 26] Solves 19.8% – 58 cases 87.5% solved – 288 positions F2L + edges F2L + all placed 70.4% solved – 663,552 positions PEO-CPLL [1] 87.2% solved – 324 positions 88.1% solved – 216 positions Solves 7.0% – 2 cases ⟨R,L,F,B,U,D⟩ =⟨R,L,F,B,U⟩R2,L2,F2,U2⟩ =⟨R,L,F,B,U⟩ [1 p. 26] ⟨R,L,F,B,U,D⟩ =⟨R,L,F,B,U⟩R2,L2,F2,B2⟩ = 3 3= 1⟨R,L,F,B,U⟩×1×1cubegroup cube group Two 1×1×1 2 3 blocks + corners + edges oriented cubegroup×1×1† Everything except a slice 73.4% solved – 165,888 positions 88.4% solved – 192 positions 83.9% solved – 1,440 positions F2L + edges oriented + corners placed PO-CPLL PEO-EPCOLL = OC(P)ELL [1] F2L + corners 86.8% solved – 384 positions 87.2% solved – 324 positions Solves 7.03% – 2 cases Solves 14.3% – 40 cases 89.9% solved – 96 positions ⟨R,L,F,B,U,D⟩ =⟨R,L,F,B,U⟩Ra,Fa,Ua⟩ =⟨R,L,F,B,U⟩ = antislice group [1 p. 11] 80.7% solved – 6,144 positions

ZBLL [1][2] Solves 19.8% – 493 cases PEOCP-COLL [1] PE-CPLL ⟨R,L,F,B,U,D⟩ =⟨R,L,F,B,U⟩R2,F2,U2⟩ =⟨R,L,F,B,U⟩ [1 p. 26] ⟨R,L,F,B,U,D⟩ =⟨R,L,F,B,U⟩R2,L2,F2⟩ =⟨R,L,F,B,U⟩ = slice-squared group [1 p. 11] Kociemba II Solves 7.3% – 7 cases Solves 5.5% – 4 cases 82.6% solved – 2,592 positions 89.9% solved – 96 positions Solves 52.4% PE-COLL = OCLL-EPP [1] PEO-CLL = COLL [1] J-EOCLL = OLLCP [1] PC-EOLL Solves 7.3% – 7 cases F-ALL = 1LLL [1] Thistlethwaite IV Solves 14.3% – 42 cases Solves 18.9% – 331 cases Solves 4.6% – 3 cases F2L + edges + corners placed Solves 24.4% – 3915 cases Solves 29.6% PO-EPLL 92.7% solved – 27 positions Two opposite layers + edges oriented † F-PLL = PLL [1] Solves 7.0% – 2 cases ⟨R,L,F,B,U,D⟩ =⟨R,L,F,B,U⟩R2,F2⟩ =⟨R,L,F,B,U⟩ = two squares group [1 p. 11] 91.4% solved – 48 positions Solves 12.5% – 21 cases 94.5% solved – 12 positions F2L + edges oriented + corners F2L + edges + corners oriented F-CLL = L4C [1] F-OLL = Pure OLL [1] 94.5% solved – 12 positions 94.5% solved – 12 positions Solves 12.8% – 84 cases Solves 11.9% – 57 cases

F-ELL = ELL [1] F-COLL = Pure CO [1] F-EPLL = EPLL [1][2] F-CPLL = CPLL [1] Solves 10.1% – 29 cases Solves 7.3% – 7 cases Permuted centers † = quotient of Void Cube group Solves 5.50% – 4 cases Solves 5.5% – 4 cases 94.5% solved – 12 positions

Solved = ⟨R,L,F,B,U,D⟩ empty = 1 1 1=⟨R,L,F,B,U⟩⟩×1×1cubegroup cube group [1] 100% solved – 1 position

WHAT IS THIS? NAMING OF LAST-LAYER STEPS MEASURING COMPLEXITY GROUP THEORY OF THE CUBE LEGEND AND NOTES

This chart gives an overview of methods to solve Rubik's Cube, breaking The number of possible steps after the first two layers have been J-XLL: The just steps. These steps solve just the given aspects and don't The total number of positions of Rubik's Cube is The methods to solve Rubik's Cube can be analysed by using the powerful A cube state, a group down each method into individual steps, and showing their relations. solved is large, and their naming can be confusing. In this chart, we preserve anything else. They are used as the first step after solving the first State A mathematical tools of group theory. use the following naming scheme, in addition to indicating traditional two layers. For instance, J-COLL is the step that orients the corners, but 8! 12! ½ 3 23⁷ 2¹¹=⁷¹¹ = 43;252,003;274,489;856,000 A step, a subgroup relationship Each dark gray rectangle represents one state of the cube, from fully names of the steps. ignores (i.e., may scramble) the edges and the permutation of corners. Individual positions of the cube can be identified with sequences of moves that lead scrambled (top) to fully solved (bottom). In between, various states of partial I.e., there are 43 trillion positions on the long scale (a trillion is from the solved cube to the particular position. In this way, any two positions can be A named step, a named subgroup relationship order in the cube's arrangement represent the stepping stones used to solve Names include the following symbols: E – Solve all edges. C – Solve PY-XLL: The preserve steps. These steps solve aspect X while a million cubed). This can be derived as: combined by considering the position reached by starting with a solved cube, and the cube. Vertical lines connect these states and represent individual steps to all corners. P – Solve all permutations. O – Solve all orientations. preserving aspect Y. These are used in the middle of a last layer method – State B executing the two sequences of moves. This binary operation is a group operation: ⟨R,L,F,B,U,D⟩ =⟨R,L,F,B,U⟩…⟩ Group generator notation: The set of all be performed. Rounded boxes give information about each step. Colored These can be combined, e.g., EO – Solve all edge orientations. they are thus only used in methods that need at least three steps for the 8! – Number of ways to permute the 8 corner pieces positions reachable using only the given lines connect the whole from top to bottom to show individual methods that last layer. For instance, PE-COLL is the step that orients corners while 12! – Number of ways to permute the 12 edge pieces • The solved position (which we call I) serves as a neutral element, because moves ½ 3⁷2¹¹= † solve the cube. There are three types of steps: preserving edges, but may scramble the corner permutation. Subset of – Factor given by the fact that the overall AI = IA = A for all A; Only in these groups is the position of the Step X Step Y permutation of the Cube's pieces is always even • Every position A has an inverse A', reached by executing all moves of A in reverse, centers not fixed Equivalently, this chart shows mathematical groups related to the Cube, and F-XLL: The finish steps. These steps finish the cube to the solved Subset relationships. In some cases, the algorithms of certain steps are a 3⁷ 2¹¹= – Number of orientations of all corners (one less that such that AA' = A'A = I; [n] Reference (HTTP link) their subgroup relationships. (See the inset about the mathematics.) state, and thus they preserve everything that they don't solve. For subset of the algorithms for another step. This is the case when one step is the number of corners because the orientation of • The operation is associative: (AB)C = A(BC) for any A, B, and C. [n p. N] Reference with page number (HTTP link) instance, F-EOLL is the step that solves the edge orientations on the broken down in several smaller steps, as shown in the diagram on the right. seven corners determines the orientation of the The chart includes methods used for (like CFOP), historically last layer, while preserving all corners and the edge permutation. In that case, the algorithms for step Y are a subset of the algorithms for last corner) For the complete scrambled state of the Cube, we have a group with “Orient” is used when permutations can be done; “rotate" important methods (like the one by Singmaster), as well as methods used by step X. State C 2¹¹ – Number of orientations of all edges (same 43;252,003;274,489;856,000 elements, also called the Rubik's Cube group [1], when not. The adjusting of the upper face (AUF) is not computers (like the Kociemba method). reasoning applies) represented at the top of the chart. This group has subgroups that correspond to mentioned; it is integrated into the first step that performs a individual partially solved states of the Cube; these subgroups are represented as permutation, for those methods that solve the first two Size of groups. Similar considerations can be used to derive rectangles in the chart. In particular, the completed state (at the bottom of the chart) layers. the number of positions in each partially solved state of the represents the trivial subgroup, which contains only the position I. Other subgroups Cube. For instance, the U group (everything solved except the serve as stepping stones for solving the cube. The chart shows the canonical steps taken by each method – U layer) has the following number of positions: but note that individual methods allow for more variations Notation. The notation for moves in Rubik's Cube differs slightly from the usual and optimizations in many cases. 4! 4! ½ 3 23⁷ 2¹¹=³2³62,208 = 62,208 mathematical notation in group theory. The notation is largely that of a multiplicative group, but inverses are notated with an apostrophe (e.g., U' rather than U−11), and The chart is already very dense as it is, and thus we refrain Solving percentage. Additionally, we indicate in the chart in powers are notated with non-superscript numbers (e.g., U2 rather than U²). ). from including additional, less notable, states and steps. percent how much of the Cube is solved at each step, using a logarithmic scale. For instance, in the U group, there are only Subgroup relationships. In the chart, each vertical line represents a subgroup HOW TO READ CUBE DIAGRAMS 62,208 positions possible, and thus the Cube is considered to be relationship. In most cases, these are not normal subgroup relationships. If the solved to relationship is one of a normal subgroup, then we can consider the quotient group, representing the subpuzzle going from one to the other. In some cases these 1 −1 log(62,208) / log(43;252,003;274,489;856,000) = 75.6% subpuzzles can be built as their own mechanical puzzle – see the insert on the left. The Western color scheme is used: white opposite yellow, red opposite PUZZLES IN A PUZZLE Note however that the trivial group is a normal subgroup of all groups, with quotient orange, and blue opposite green. Gray represents any color. Other colors Note that this measure does not take into account the fact that corresponding to each group. (pink, turquoise, brown) represent groups of pieces permuted and oriented in it becomes more and more difficult to solve the cube the more specific ways (the details are given in the group name or in the references). The Rubik's Cube is not the only twisty puzzle; other puzzles of similar construction exist. In some cases, these puzzles appear as subpuzzles of the pieces are already placed, meaning that the same percentage at Uniqueness of subgroups. Note that subgroups are not unique. For instance, the the bottom of the chart needs longer algorithms in general than U group (F2L is solved) can be embedded in multiple ways into the overall Rubik's Not all groups can be readily visualized using such a diagram: In some cases, Rubik's Cube. For instance, by turning only the front, back, left and right faces by half turns, the Rubik's Cube is reduced to a puzzle that is exactly equivalent to the 3×1×1 3 1cubegroup×1×1 puzzle shown below. In similar ways, a whole variety of twisty puzzle is found within the Rubik's cube. at the top of the chart. Nonetheless, this number gives a rough cube group – one for each way to rotate the cube, giving 24 in total. The size of each we show an arbitrary position from the group, while some groups don't have a estimate of how much each step solves. group (as shown in each rectangle) gives the number of positions for each choice of cube diagram shown. Mathematically, these represent subgroups of the Rubik's Cube group. Inversely, the Rubik's Cube appears as a subpuzzle of larger puzzles, such ABOUT as the 4×1×1 4 cubegroup×1×14 cube. embedding. Number of cases. For steps that can be broken down into a list of algorithms, we also give the number of such algorithms Commutators. Another way that group theory is used is by the use of commutators Pieces drawn in one of When center facets are that have to be learned. For instance, the case F-PLL, i.e., – specific ways to combine two algorithms to a new one. As an example, commutators Chart created by Jerome Kunegis, 2019. Main references: the six cube colors are gray, the center pieces permuting the last layer, which is the last step in the CFOP are used in the method labeled Spiegel for the last three steps. Given two positions A SpeedSolving.com Wiki (https://www.speedsolving.com/ fixed; pieces in gray are scrambled too (this method, has 21 different cases. This count always considers and B, their commutator is defined as [A, B] = ABA'B'. The set of all commutators wiki/index.php/Main_Page) • Notes on Rubik's Magic can have any position is also shown as “† ”) mirrorings and inversions as different cases. If those are forms a subgroup, which for the Rubik's Cube contains half of all positions – indicating Cube (https://maths-people.anu.edu.au/~burkej/cube/ The 2×1×1 2cubegroup cube, also known counted as the same, the number will be less. that the Rubik's Cube group is highly non-commutative, making it a difficult puzzle. singmaster.pdf) • Wikipedia: Rubik's Cube group (https:// as the , en.wikipedia.org/wiki/Rubik%27s_Cube_group) appears as a subpuzzle Generators. Certain subgroups can be represented using generators. These are when going from the indicated with angle brackets ⟨R,L,F,B,U,D⟩ .=⟨R,L,F,B,U⟩…⟩ For instance, ⟨R,L,F,B,U,D⟩R2,L2,F2,B2 =⟨R,L,F,B,U⟩⟩ represents the set of Image credits: Pocket Cube, Wikimedia Commons [1] • scrambled state to the “all The 1×2 1 1method)×2 cube corresponds all positions that can be reached by half turns of the left, right, front and back faces to the trivial group. It can be Floppy Cube, Wikimedia Commons [2] • Void Cube, Lazada If only one facet of Pieces drawn in other corners solved (but not the The Void Cube is similar to the Rubik's Cube but lacks only. This restricts the positions that can be reached, and in fact only 192 positions [3] • 1 1 ×1×1 cubegroup×1×11 Cube, eBay [4] centers)” state. It can be emulated using a solved Rubik's can be reached in this way, corresponding exactly to all positions that a 3×1×1 3 1cubegroup×1×1 several pieces is non- colors have fixed orien- centers. As a result, its group corresponds to the Cube and not performing any gray, then the permu- tation and are permu- emulated using a Rubik's The 3×1×1 3 cubegroup×1×11, also known as the quotient of the Rubik's Cube group with the “all is Floppy Cube can be in. I have tried to include the most important methods, but had Cube by ignoring all centers and edges. It has Floppy Cube, appears as a subpuzzle turns. Its main use is as a tation of those pieces ted in specific ways in solved but centers may be permuted" group. It can be running gag in the cubing to make choices due to space constraints. The Spiegel is fixed each color group 3;674,160 positions. It is more complex to solve by only executing half turns of the emulated by ignoring the centers of a Rubik's Cube, but method was originally published in the German magazine than one might at first think, and serious R/L/F/B sides. It has 192 positions community, as well as a useful that could interpreted as cheating because the position addition to any collection of and is included because it was the first method I learned. It speedsolving competitions are held for it – the and can be solved easily by most of centers may help the solution. It has 12 times less is based on 's method. world record stands at less than one second. people. twisty puzzles, even though it is, positions than the Rubik's Cube. strictly speaking, not twisty.