Group Theory of the Cube Measuring

Extended Chart of Methods for Solving Rubik's Cube .1/ CFOP/Fridrich .1/ Petrus CFCE .1/ Sin#! ster .1 p. 61/ Roux .1/ Zborowski .1/ Tripod .1/ .1/ Zborowski–*ruche! S+iege( .1/ W ter! " .1] .1/ EG $%×2 method)2 method& .1/.%/ Orte# Guimond (2×2 2method) method& .1] .1/ Thistlethw ite .1/ Kocie!b &cram#led = ⟨R,L,F,B,U,D⟩'()(*(+(,(- = '()(*(+(, =⟨R,L,F,B,U⟩⟩⟨R,L,F,B,U,D⟩ .1] 0% solved – 43;%$%(0012%34,4892856,000 positions %×1×1 % 2cubegroup×1×1 #lock .1/ &olves %3.5% *our corners "ross &olves %0.6% *our corners oriented † G7)ine %×1×1 % 2 block = '(*(, cubegroup×1×1⟨R,L,F,B,U,D⟩=⟨R,L,F,B,U⟩⟩ .1 p. %4/.%/ &olves 26.9% 13.6% solved – 91,538;431,%48;761(600 positions :ll corners oriented T;istlet;waite J &olves 27.3% 27.5% solved – 1702659,31$2142,400 positions 17.0% solved – 19,3342864(780;288,000 positions &olves 16.9% 1×1×1 % 3 block cubegroup×1×1† .1] 31.2% solved – 1%2$04(416;213(400 positions %×1×1 % 3cubegroup×1×1 #lock .1] *our corners † 7rient remaining our corners ⟨R,L,F,B,U,D⟩ =⟨R,L,F,B,U⟩commutator⟩ =⟨R,L,F,B,U⟩ = doubly even permutation .1 p. %3/ &olves 19.3% "ross .1] %0.6% solved – 1(814;109,4142198,400 positions &olves 7.3% – 7 cases 1.5% solved – 21;4%4(001;413(244;928(000 positions 26.9% solved – %%32546,313;$%1(%00 positions Kociem#a J %×1×1 % cubegroup×1×13 block .1] G< .1] G>change &olves 47.6% 46.8% solved – 28,2132548(800 positions "ross + one pair .1/ 7ne layer .1] &olves 21.3% – 128 cases "8)) 7)) %×1×1 %cubegroup .1] &olves 3.9% – 4 cases ⟨R,L,F,B,U,D⟩ =⟨R,L,F,B,U⟩'()(*(+(,%(-%⟩ =⟨R,L,F,B,U⟩ 40.0% solved – 592,568;$24,800 positions $3.0% solved – 1,43%2151,040 positions &olves 14.3% – 42 cases &olves 7.1% – 7 cases 16.9% solved – 21,119;14%(%%12872,000 positions )ine + edges oriented = G7)ine = ⟨R,L,F,B,U,D⟩'()(* =⟨R,L,F,B,U⟩⟩ .1 p. %4/.%/ 7rient edges .1/ :ll corners oriented in t;eir ,I- layer † 27.7% solved – 1592993,$01;694(000 positions &olves 9.%% "ross + two adjacent pairs .1] "ross + two opposite pairs *our middle slices 20.9% solved – 1(190;319,43424%0(800 positions 52.6% solved – %(01$2539(200 positions 52.6% solved – %(01$2539(200 positions &olves 22.6% T;istlethwaite JJ Two 1×1×1 % 1 #loc9s cubegroup×1×1† &olves 30.7% 61.9% solved – 29;859,840 positions %×1×1 % cubegroup×1×13 block + edges oriented "ross + t;ree pairs .1/ *our corners + ot;er corners oriented † $6.0% solved – 4402899(200 positions 44.5% solved – 9;331,200 positions 27.9% solved – 141;%41(319;859,%00 positions L<)& *%) = , group .1] P+) .1/ &olves %1.0% 75.6% solved – 4%(%08 positions P)) %×1×1 %cubegroup .1] &olves 14.1% – 5 cases HH *%) %×1×1 1 1cubegroup×1×1 #lock .1] &olves 7.0% – 2 cases &olves 52.$% &olves 24.%% G7)& .1] 56G7)) .1/ 56G)) = ))G* .1] 56P)) ⟨R,L,F,B,U,D⟩ =⟨R,L,F,B,U⟩'()(*%(+%(,%(-%⟩ =⟨R,L,F,B,U⟩ "L)) .1] &olves 15.3% – 102 cases &olves 4.40% – 3 cases &olves 11.6% – 15 cases &olves 12.5% – 21 cases 47.6% solved – 19(508;428,800 positions &olves 14.3% – 42 cases Tripod ⟨R,L,F,B,U,D⟩ =⟨R,L,F,B,U⟩'(*⟩ =⟨R,L,F,B,U⟩ .1 p. $$/ 56"7)) 80.2% solved – 3(336 positions $9.9% solved – 312483,200 positions &olves 7.%9% – 7 cases 56")) = ")) .1/ 56GP)) "orners † = quotient o % ×1×1 % 2cubegroup×1×1 cube group Gdges placed = %×1×1 % 2cubegroup×1×1 cube group + even rotations "PG7)) .1] &olves 14.3% – 42 cases &olves 7.0% – 2 cases 35.0% solved – $2885(971;440(800 positions 61.1% solved – 44;089,920 positions &olves 11.6% – 15 cases ⟨R,L,F,B,U,D⟩ =⟨R,L,F,B,U⟩L(G(&('a,*a,,a⟩ =⟨R,L,F,B,U⟩ .1 p. 14/ *%) + edges oriented .1] *%) + corners oriented *%) + edges placed 43.3% solved – 15;92$(248 positions 80.2% solved – 3(376 positions 82.9% solved –2(304 positions 82.6% solved – %($92 positions 7ne layer + corners T;istlethwaite JJJ Two 1×1×1 % 3 #loc9s + our corners cubegroup×1×1† PGP6G7)) 47.3% solved – %2580(480 positions &olves 22.8% 76.3% solved – 46,080 positions ⟨R,L,F,B,U,D⟩ =⟨R,L,F,B,U⟩'(*%⟩ =⟨R,L,F,B,U⟩ .1 p. $3/ PG76"7)) = 7")) .1] P"76G7)) = 7G)) .1] &olves 4.6% – 3 cases 78.8% solved – 14,400 positions &olves 7.%9% – 7 cases &olves 4.40% – 3 cases 567)) = 7)) .1/ &olve 11.9% – $3 cases P"76"P)) PG76GP)) ⟨R,L,F,B,U,D⟩ =⟨R,L,F,B,U⟩L(G(&⟩ = slice group =⟨R,L,F,B,U⟩† .1 p. 10/ &olves 7.0% – 2 cases &olves 7.0% – 2 cases 85.3% solved – 348 positions Tripod last step .1] *%) + oriented .1] ⟨R,L,F,B,U,D⟩ =⟨R,L,F,B,U⟩'%()%(*%(+%(,%(-%⟩ = '%()%(*%(+%(,%= ⟨R,L,F,B,U⟩⟨R,L,F,B,U,D⟩=⟩ = square group .1 p. %4/ &olves 19.8% – 58 cases 87.5% solved – 288 positions *%) + edges *%) + all placed 30.4% solved – 441($52 positions PG76"P)) .1] 87.2% solved – 324 positions 88.1% solved – 216 positions &olves 7.0% – 2 cases ⟨R,L,F,B,U,D⟩ =⟨R,L,F,B,U⟩'%()%(*%(,%⟩ =⟨R,L,F,B,U⟩ .1 p. %4/ ⟨R,L,F,B,U,D⟩ =⟨R,L,F,B,U⟩'%()%(*%(+%⟩ = 1 1= 1⟨R,L,F,B,U⟩×1×1cubegroup cube group Two 1×1×1 % 1 #loc9s + corners + edges oriented cubegroup×1×1† Gverything except a slice 33.4% solved – 16$(888 positions 88.4% solved – 192 positions 83.9% solved – 1,440 positions *%) + edges oriented + corners placed P76"P)) PG76GP"7)) = 7"?PAG)) .1] *%) + corners 86.8% solved – 384 positions 87.2% solved – 1%4 positions &olves 7.03% – 2 cases &olves 14.3% – 40 cases 89.9% solved – 96 positions ⟨R,L,F,B,U,D⟩ =⟨R,L,F,B,U⟩'a,*a,,a⟩ =⟨R,L,F,B,U⟩ = antislice group .1 p. 11] 80.7% solved – 4(144 positions H+)) .1].%/ &olves 19.8% – 493 cases PG7"P6"7)) .1] PG6"P)) ⟨R,L,F,B,U,D⟩ =⟨R,L,F,B,U⟩'%(*%(,%⟩ =⟨R,L,F,B,U⟩ .1 p. %4/ ⟨R,L,F,B,U,D⟩ =⟨R,L,F,B,U⟩'%()%(*%⟩ =⟨R,L,F,B,U⟩ = slice-squared group .1 p. 11] Kociem#a JJ &olves 7.3% – 7 cases &olves 5.$% – 4 cases 82.6% solved – %(592 positions 89.9% solved – 96 positions &olves 52.4% PG6"7)) = 7"))6GPP .1] PG76")) = "7)) .1] 56G7")) = 7))"P .1] P"6G7)) &olves 7.3% – 7 cases *6:)) = 1L)) .1] T;istlethwaite J! &olves 14.3% – 42 cases &olves 18.9% – 111 cases &olves 4.6% – 3 cases *%) + edges + corners placed &olves 24.4% – 1915 cases &olves 29.4% P76GP)) 92.7% solved – 27 positions Two opposite layers + edges oriented † *6P)) = P)) .1] &olves 7.0% – 2 cases ⟨R,L,F,B,U,D⟩ =⟨R,L,F,B,U⟩'%(*%⟩ =⟨R,L,F,B,U⟩ = two squares group .1 p. 11/ 91.4% solved – 48 positions &olves 12.5% – 21 cases 94.5% solved – 12 positions *%) + edges oriented + corners *%) + edges + corners oriented *6")) = )4C .1] *67)) = Pure 7)) .1/ 94.5% solved – 12 positions 94.5% solved – 12 positions &olves 12.8% – 84 cases &olves 11.9% – 57 cases *6G)) = G)) .1/ *6"7)) = Pure "O .1] *6GP)) = GP)) .1/.%/ *6"P)) = "P)) .1/ &olves 10.1% – %9 cases &olves 7.3% – 7 cases Permuted centers † = quotient of !oid "u#e group &olves 5.$0% – 4 cases &olves 5.5% – 4 cases 94.5% solved – 12 positions &olved = ⟨R,L,F,B,U,D⟩empty = 1 1 1=⟨R,L,F,B,U⟩⟩×1×1cubegroup cube group .1/ 100% solved – 1 position WHAT .S T,.S0 NA/.NG OF LAST-LAYER STEPS /EAS-R.NG CO/PLE6.TY GRO-P T,EORY OF THE C-BE LEGEND AND NOTES This c;art gives an overview of methods to solve 'ubi9Os "ube( brea9ing The number o possible steps a ter the irst two layers ;ave #een 56T))= T;e <ust steps. These steps solve just the given aspects and donOt T;e total number of positions of Rubi9Os Cube is T;e met;ods to solve 'ubi9Os "ube can #e analysed #y using the powerful : cube state( a group down eac; met;od into individual steps, and s;owing their relations. solved is large( and their naming can #e confusing. Jn this c;art( we preserve anyt;ing else. They are used as t;e first step a ter solving the first &tate : mat;ematical tools of grou+ theor2. use the following naming sc;eme( in addition to indicating traditional two layers. *or instance( 56"7LL is the step that orients the corners( but 8! 12! ½ 3 23⁷ 2¹¹=⁷¹¹ = 43;%$%(0012274,489;856,000 : step, a subgroup relationship Gach dar9 gray rectangle represents one state of the cube( rom ully names of the steps. ignores (i.e.( may scram#leA t;e edges and the permutation of corners.

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