Thistlethwaite's 52-move algorithm

Morwen B. Thistlethwaite is a mathematician who devised a clever algorithm for solving the Rubik's Cube in remarkably few moves. It is a rather complicated method, and therefore cannot be memorised. It is only practical for computers and not for humans. This algorithm is rather important from a theoretical standpoint however, as it has long been the method with the fewest number of moves.

Thistlethwaite's method differs from layer algorithms and corners first algorithms in that it does not place pieces in their correct positions one by one. Instead it works on all the pieces at the same time, restricting them to fewer and fewer possibilities until there is only one possible position left for each piece and the cube is solved.

The way it does this is by first doing a few moves until a position arises that can be solved without using quarter turns of the U and D faces (though half turns of U and D are still needed). It then proceeds to solve the cube without using U or D quarter turns by first moving to a position that does not need quarter turns of the F, B faces either. With these further restrictions a position is arrived at that does not need any quarter turns at all, and can hence be solved by half turns only. The cube then indeed gets solved using half turns only.

These four stages are quite complicated, since they use large look-up tables for all the positions at each stage. The table below shows the numbers involved:
Group# positions  Factor
G0=<L, R, F, B, U, D>4.33·1019
2,048(211)
G1=<L, R, F, B, U2,D2>2.11·1016
1,082,565(12C4 ·37)
G2=<L, R, F2,B2,U2,D2>1.95·1010
29,400( 8C42 ·2·3)
G3=<L2,R2,F2,B2,U2,D2>6.63·105
663,552(4!5/12)
G4=I1

Mathematically speaking, this is a sequence of nested groups, and each stage of the algorithm is simply a look-up table showing a solution for each element in the quotient coset space. The last column shows the order of these coset spaces, i.e. the size of the look-up tables used.

The first stage has a factor of 2,048=211. This corresponds to the fact that this stage fixes the orientation of the edges; it is impossible to flip edges if only half turns are allowed on the U, D faces, but otherwise the pieces can be put in any normal position with this restriction.

The second stage has a factor of 37 *12!/(8!4!). This corresponds to the fact that this stage fixes the orientation of the corners, and places the middle layer edges into their slice. It is fairly easy to see that if you are only allowed half turns on the F,B,U,D faces together with any turns on the L, R faces, then the left and right faces will never have more than two colours, and the middle edge pieces will never leave their slice or be flipped. In fact, those restricted moves can solve any position under those conditions.

The third stage has a factor of [8!/(4!4!)]2 *2*3. This corresponds to the fact that the edges in the L and R faces are placed in their correct slices, the corners are put into their correct tetrads, the parity of the edge permutation (and hence the corners too) is made even, and the total twist of each tetrad is fixed.

The final stage finally solves the cube.

The following table shows the number of moves each stage needs.
1234Total
Original algorithm:713151752
Improved algorithm:713151550
Best possible:710131545

To keep the size of the lookup tables down, Thistlethwaite used simplifying preliminary moves which is why the number of moves he needed is sometimes more than the best possible (the 'diameter' of the quotient spaces). His students first improved the algorithm by completely analysing the square group, thus reducing it to 50 moves. Later each stage was fully computed, for a 45 move solution. In 1991 Hans Kloosterman reduced that to 42 moves by using slightly different stages 3 and 4.

Through extensive calculations, Mike Reid has shown that going from G0 directly to G2 can be done in 12 moves, and from G2 to G4 in 18 moves. When combining these, the 30-move cases can be avoided, so it has been proved that 29 moves is always sufficient to solve the cube. More recently other techniques have allowed computers to verify that God's Algorithm is 20 moves in the worst case.

Ryan Heise has developed a way of solving the cube based on Thistlethwaite's algorithm. He splits stages 2, 3 and 4 into two steps each (corners and edges separately) in order to make it possible for a human being to memorise.

David Singmaster has made scans of Thistlethwaite's printouts, and I converted them to text. Below is a reproduction of those papers, as accurate as I can make them. It consists of
1. Covering letter, 1 page
2-4. General Instructions, 3 pages
5-10. Stage 2 tables, 6 pages
11-18. Stage 3 tables, 8 pages
19-25. Stage 4 tables, 7 pages
26. Detailed example, 1 page.

Note that the hand-written corrections in the introduction are shown in italics here.

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26 Queens Road, Loughton, Essex. 13 July 1981. Dear Thank you for your letter asking for a copy of my 52-move strategy for solving Rubik's Cube. Please find enclosed the following:- (i) General instructions and description (3 pages); (ii) Tables for Stage 2 (computer print-out); (iii) Instructions and table for the first part of Stage 3; (iv) Tables for the latter part of stage 3 (7 typewritten pages); (v) Tables for Stage 4 (7 pages of computer print-out, photocopied). I should perhaps point out that this strategy isn't easy to perform (even John Conway finds it quite hard!), for two reasons: first, only one representative of each symmetry class is given in the tables; second, I have given only the barest documentation. I hope that a fuller description will be forthcoming at some stage. On reading the bottom of page 2 of the General Instructions, you will note that in fact Stage 2 requires two sets of tables. I have omitted to send you the second of these, because they are bulky, and only save one move! I hope you will excuse this liberty. One can always get FU,FD,BU,BD into the UD-slice (the middle horizontal slice) in at most five moves in G1. As an example of the strategy at work, consider the position where the four upper corners are twisted clockwise, the four lower corners are twisted antlclockwise, and all twelve edges are flipped. Then the cube is restored by:- Stage 1 DBFUR'L'D Stage 2 L2F2D2F | L2R2F'R'BR2B'R'B' Stage 3 L2B2 | U2LU2F2L2F2L'B2LB2L' Stage 4 R2U2 | R2D2F2U2B2D2F2B2L2B2L2B2 Yours sincerely, Morwen B. Thistlethwaite.

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page 1 THE 45-52 MOVE STRATEGY Introduction. Let G= <L,R,F,B,U,D> , G1= <L,R,F,B,U2,D2> , G2= <L,R,F2,B2,U2,D2> , G3= <L2,R2,F2,B2,U2,D2> . The plan is to manoeuvre down through the chain G=G0> G1> G2> G3> 1. One gets from Gi to Gi+1 by using moves in Gi only. In its shortest form this strategy would be executed with the help of a computer, in which case I conjecture only 45 moves would be needed, but here I have sacrificed 7 moves in order that there should be no need for a computer. With a few more pages of tables, the figure 52 could be reduced to an intermediate figure of, say, 49. I intend to do this shortly! The indexes of the chain of subgroups are 2048, 1082565, 29400, 663552, but these figures are considerably reduced by considering symmetries. The reader may check that these indexes multiply together to give the order of G. The accompanying tables are broadly classified according to corner positions, and in detail according to edge positions. The words listed actually produce the positions under consideration, so the restoring moves are the inverses of these. In order to be able to use the tables it is necessary to understand the basic characteristics of the groups G1, G2, G3, so the necessary facts are presented below. Getting into G1. This involves edge pieces only, and is easy, for which reason no tables are given. An edge piece is BAD if in taking it home an odd number of quarter-turns of U and D faces is needed; otherwise it is GOOD (note that badness is well-defined). The reader may quickly work out a rule of thumb for deciding whether a piece is GOOD or BAD. Now quarter-turns of either U or D faces convert BAD pieces to GOOD and vice versa; other moves have no effect. Therefore to make all edge pieces GOOD, move groups of them to U or D face avoiding quarter-turns of U or D, and then cure them by performing a quarter-turn of U or D. For example, if all twelve are BAD, DBFUR'L'D will cure them all!

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page 2 Getting from G1 into G2. What has been achieved so far (although it doesn't look like it) is the correct orientation of edge pieces! In the present stage, the same is accomplished for corners, and also the edge pieces FU, FD, BU, BD are brought into their slice. As is well known, corners do not in general have a natural orientation, but here, roughly speaking, we shall line them all up the same way. More precisely, note that each corner piece has either a L-facet or a R-facet: on completion of this stage each of these facets will lie on either the L or the R face. In fact, the same will be true of the eight edge pieces with L or R facets, in view of the statement above regarding FU, FD, BU, BD. There are 1082565 cases to consider here, this number being the product of 3^7(total number of corner orientations) , and 12C4 (total number of arrangements of the set {FU, FD, BU, BD} amongst the twelve edge positions). Surprisingly, with a certain amount of practice it is possible to get through this stage in at most 17 moves without tables; the same is most certainly not true of the next stage although there are only 29400 cases. However, with a few pages of tables this figure of 17 may be reduced to 13; with a great deal more computation, it should be possible to reduce it further to 10. To "prove" that 10 moves were sufficient, one would run through all 10-move sequences on the computer, and check that 1082565 different cases resulted. This would take no more than a few hours of computer time, in view of certain short cuts available by considering symmetries. Now to business. The twist of a corner is measured by looking at its L or R facet and observing how this has been rotated in relation to the adjacent L or R face. Note that quarter-turns of F and B faces alter the twist of corners, whereas all other moves in the group G1 have no effect. Now in at most 4 moves in G1 try to obtain a position where either the corners on the F face or the corners on the B face have zero twist. Note the twist of the corners on the opposite face and also the positiosn of the edge pieces the edge pieces FU, FD, BU, BD are all in the UD-slice. If this is not possible, get them all in the U-face in at most 4 moves.

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page 3 FU, FB, BU, BD, and tThen refer to the appropriate detailed table. The words listed in these tables need to be inverted, as mentioned earlier. Getting from G2 into G3. This is the trickiest stage theoretically, and may be broken down for purposes of clarification into two sub-stages: first get corners into their natural orbits, and second permute the corners within their orbits so as to obtain one of the 96 corner permutations in the squares group (G3), while at the same time sorting out the edge pieces into their correct slices. The table of initial moves on the first page of Stage 3 tables does part of the first of these substages, and the detailed tables do the rest. After performing the initial moves, the set of corners out of orbit will be one of three possibilities (modulo symmetries): (i) the empty set; (ii) the set {1, 8, 2, 7} ; (iii) the set {1, 5} . The reader then has to calculate which coset of form G3αβ the permutation of corners lies in, where α is one of 1, (14)(68), (24)(58), (12), (14), (24), and β is one of 1, (18)(27), (15). Since some of these cosets are reflections of others, it was not necessary to produce tables for all six possible values of α. The task of reflecting positions if necessary is left to the reader. I apologise for the slightly anomalous numbering of edge positions for this stage; this was due originally to a typing error in a programs and has stuck ever since! Getting from G3 to home. In this final stage one uses only 180 turns. The order of G3 is 96x6912 = 663552. The tables for this stage give words for producing each of the 6912 edge positions with corners fixed. Therefore one must first restore the corners (in at most 4 moves), and then use the tables to restore the edges. Considerable practice is needed to use these tables efficiently, but I have found (after considerable practice) that I can find the desired move in about 2 minutes. One hint is that when faced with three 3-cycles, consider the configuration of the fixed pieces. Also it pays to get to know the different sorts of 4-cycle.

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STAGE 2 TABLES FIRST BATCH OF NUMBERS: THE MOVES. 11=L 12=L2 13=L' 21=F 22=F2 23=F' 31=R 32=R2 33=R' 41=B 42=B2 43=B' 51=U2 52=D2 SECOND BATCH OF NUMBERS: TWISTS OF CORNERS. 1=CLOCKWISE 2=ANTICLOCKWISE 0=NO EFFECT FINAL NUMBER: NUMBER OF SYMMETRICALLY EQUIVALENT POSITIONS. N.B. FOR INITIAL 1X READ 21 1X. 21 11 31 22 11 31 51 52 21 0 0 0 0 0 0 0 0 1 21 11 33 42 11 51 33 52 21 0 0 0 1 0 0 2 0 4 21 11 22 51 52 12 22 33 21 0 0 0 1 1 0 2 2 8 21 11 22 31 43 51 13 22 11 0 0 0 1 1 1 2 1 8 21 11 52 11 21 31 52 41 51 0 0 0 1 2 2 0 1 8 21 11 22 12 52 12 52 31 41 0 0 0 2 0 1 2 1 8 11 21 33 23 11 32 51 52 23 0 0 1 0 0 0 2 0 4 11 21 33 23 11 32 51 52 21 0 0 1 0 2 0 0 0 4 11 42 31 21 13 42 33 41 51 0 0 1 1 0 0 2 2 2 21 11 21 33 42 11 23 12 43 0 0 1 1 0 1 0 0 8 21 11 22 31 41 11 32 42 13 0 0 1 1 1 0 1 2 8 21 11 31 21 51 31 21 41 11 0 0 1 1 2 2 0 0 2 21 11 22 32 41 51 13 22 11 0 0 1 1 2 2 2 1 8 21 11 21 41 11 23 41 11 43 0 0 1 2 0 0 0 0 8 21 11 41 51 41 12 42 11 41 0 0 1 2 0 0 1 2 4 21 11 42 12 41 52 41 11 41 0 0 1 2 0 0 2 1 4 21 11 21 32 42 11 23 12 43 0 0 1 2 0 1 0 2 8 21 11 22 12 22 51 31 52 21 0 0 1 2 1 0 0 2 8 21 11 21 12 23 42 11 43 12 0 0 1 2 1 0 1 1 8 21 11 22 13 21 31 21 11 41 0 0 1 2 1 1 0 1 8 21 11 21 41 11 23 43 11 43 0 0 1 2 2 1 0 0 4 21 11 22 11 52 33 51 21 51 0 0 2 0 0 0 0 1 4 21 11 22 11 22 52 33 51 23 0 0 2 0 0 0 1 0 4 21 11 22 11 52 33 51 21 12 0 0 2 0 0 1 0 0 4 21 11 22 11 22 52 33 51 21 0 0 2 0 1 0 0 0 4 21 11 33 42 11 21 12 31 43 0 0 2 1 1 0 0 2 8 21 11 22 13 21 13 21 12 43 0 0 2 1 1 1 2 2 8 21 11 21 11 52 11 23 11 43 0 0 2 2 0 0 1 1 2 21 11 22 11 32 52 33 51 23 0 0 2 2 0 0 2 0 8 21 11 23 11 33 51 21 12 43 0 0 2 2 2 0 0 0 8 21 11 51 31 23 52 12 21 33 0 1 0 0 0 2 1 2 8 21 11 22 51 52 12 22 33 23 0 1 0 0 2 0 1 2 8 21 11 52 11 21 31 52 41 12 0 1 0 0 2 1 0 2 8 21 11 32 43 12 31 43 51 31 0 1 0 0 2 2 2 2 8 21 11 21 43 11 31 21 51 31 0 1 0 1 2 0 1 1 8 21 12 22 51 52 31 22 32 21 0 1 0 1 2 0 2 0 2 11 23 41 31 41 11 33 52 21 0 1 0 1 2 2 0 0 8 21 11 22 12 51 31 52 21 12 0 1 0 2 0 0 1 2 8 21 11 32 51 43 52 33 23 12 0 1 0 2 0 2 0 1 4 11 22 11 51 41 11 52 23 51 0 1 0 2 1 0 2 0 4 21 11 42 51 41 11 33 23 33 0 1 0 2 2 1 2 1 8

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21 11 23 31 41 33 23 52 31 2 0 1 1 2 0 2 1 8 11 21 12 22 12 31 21 13 41 2 0 1 1 2 2 0 1 4 21 11 21 12 22 33 21 11 41 2 0 1 2 0 0 1 0 8 21 11 22 12 23 51 31 52 21 2 0 1 2 0 0 2 2 8 21 11 21 11 23 13 42 13 43 2 0 1 2 0 1 0 0 8 21 11 21 51 11 51 23 12 43 2 0 1 2 0 1 1 2 4 21 11 21 42 13 43 13 23 11 2 0 1 2 0 2 2 0 8 21 11 21 32 42 31 23 13 41 2 0 1 2 1 2 1 0 8 21 11 21 42 31 23 13 43 51 2 0 1 2 2 0 1 1 8 21 11 22 11 21 42 13 32 43 2 0 1 2 2 0 2 0 8 21 11 21 43 11 23 43 11 43 2 0 1 2 2 1 0 1 8 21 11 22 33 22 42 13 43 51 2 0 1 2 2 1 1 0 4 21 12 21 43 52 21 12 32 41 2 0 2 0 0 1 0 1 2 21 11 22 42 11 52 33 51 21 2 0 2 0 1 0 1 0 2 21 11 21 12 22 13 21 12 43 2 0 2 1 0 1 1 2 8 21 11 21 12 22 13 21 12 41 2 0 2 1 0 2 1 1 8 21 11 21 12 21 43 11 22 11 2 0 2 1 1 1 1 1 8 21 11 23 33 42 51 21 13 41 2 0 2 1 2 1 0 1 4 21 11 21 12 21 32 22 11 41 2 0 2 2 0 0 1 2 8 21 11 21 12 21 32 22 11 43 2 0 2 2 0 2 1 0 8 21 11 22 11 51 41 11 52 23 2 1 0 0 0 0 2 1 4 21 11 32 21 12 52 13 51 21 2 1 0 0 2 2 0 2 8 21 11 21 12 31 23 11 41 51 2 1 0 1 1 0 1 0 8 21 11 51 23 51 43 13 22 13 2 1 0 1 2 0 2 1 4 21 11 22 31 52 23 52 13 43 2 1 0 1 2 1 0 2 8 21 11 21 11 31 23 12 41 12 2 1 0 2 0 0 0 1 8 21 11 21 32 21 13 22 11 41 2 1 0 2 1 1 2 0 8 21 11 22 51 43 11 21 42 11 2 1 1 0 0 2 1 2 8 21 11 31 21 13 42 33 43 12 2 1 1 0 0 2 2 1 4 21 11 31 21 13 42 33 41 12 2 1 1 0 1 2 2 0 4 21 11 22 32 41 13 22 11 22 2 1 1 0 2 0 1 2 4 21 11 21 11 22 11 21 11 41 2 1 1 1 0 0 0 1 8 21 11 21 32 21 41 13 22 11 2 1 1 1 1 1 2 0 8 11 22 31 21 12 31 21 11 41 2 1 1 1 2 1 2 2 4 21 11 21 11 51 21 11 41 13 2 1 1 2 0 2 0 1 8 21 11 22 12 23 11 22 11 41 2 1 1 2 1 2 2 1 2 21 11 22 42 52 23 52 13 43 2 1 1 2 2 1 0 0 8 21 11 21 42 32 23 13 22 43 2 1 2 0 1 0 1 2 4 21 11 21 42 32 23 13 22 41 2 1 2 0 1 2 1 0 4 21 11 31 21 11 31 51 52 21 2 1 2 1 0 0 0 0 4 21 11 21 13 33 43 13 23 11 2 1 2 1 0 1 2 0 8 21 11 21 13 21 41 11 21 11 2 1 2 1 0 2 0 1 8 21 11 21 13 33 43 11 23 11 2 1 2 1 1 1 1 0 8 21 11 21 13 21 43 11 21 11 2 1 2 1 1 2 0 0 8 21 11 22 51 21 51 13 43 12 2 1 2 1 1 2 1 2 2 21 11 21 51 12 32 41 13 23 2 1 2 1 1 2 2 1 8 21 11 21 11 43 13 51 11 23 2 1 2 2 1 1 1 2 4 21 11 22 51 12 41 13 52 21 2 2 0 0 0 0 1 1 2 21 11 21 12 23 11 22 41 12 2 2 0 0 0 1 0 1 8 21 11 22 32 23 33 23 13 43 2 2 0 1 0 1 1 2 8 21 11 21 33 21 32 22 13 43 2 2 0 1 0 2 1 1 4 21 11 21 13 22 41 31 23 33 2 2 0 2 1 0 1 1 4 21 11 21 11 33 52 23 12 41 2 2 0 2 2 0 1 0 8 21 11 21 32 21 32 22 13 43 2 2 0 2 2 2 1 1 8 21 11 21 51 13 21 51 41 12 2 2 1 0 0 2 1 1 4 21 11 22 32 51 21 51 13 41 2 2 1 0 2 2 1 2 8 21 11 22 13 43 51 41 33 41 2 2 1 0 2 2 2 1 8 21 11 51 21 12 22 51 33 41 2 2 1 1 0 0 0 0 4 21 11 21 12 21 51 33 51 41 2 2 1 1 0 2 0 1 8 21 11 21 12 21 11 22 13 41 2 2 1 1 1 1 0 1 8 21 11 21 33 41 33 21 52 33 2 2 1 1 1 2 1 2 8 21 11 21 12 51 33 23 51 41 2 2 1 2 0 0 1 1 8 21 11 21 11 22 31 21 12 43 2 2 1 2 0 1 1 0 8 21 11 21 32 21 33 22 13 43 2 2 1 2 0 2 1 2 8 21 11 21 11 51 11 41 13 23 2 2 1 2 0 2 2 1 8 21 11 21 32 22 33 21 13 43 2 2 1 2 2 0 1 2 8

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21 11 21 11 51 11 41 13 21 2 2 1 2 2 2 0 1 8 21 11 21 13 43 13 23 13 51 2 2 2 0 0 0 2 1 8 21 11 21 13 22 41 31 23 31 2 2 2 0 1 0 1 1 4 21 11 21 13 22 43 31 23 31 2 2 2 0 1 1 0 1 4 21 11 21 13 43 11 23 11 22 2 2 2 0 1 1 1 0 4 21 11 21 13 21 43 13 21 11 2 2 2 1 0 2 0 0 8 21 11 21 33 21 33 22 13 41 2 2 2 1 0 2 1 2 8 21 11 21 11 43 13 51 11 21 2 2 2 1 1 1 1 2 4 21 11 21 13 21 43 13 21 13 2 2 2 1 2 0 0 0 8 21 11 21 11 22 13 21 13 43 2 2 2 1 2 1 0 2 8 21 11 21 11 22 13 21 13 41 2 2 2 1 2 2 0 1 8 21 11 21 11 21 12 22 12 41 2 2 2 2 0 0 0 1 8 21 11 21 13 23 31 21 33 41 2 2 2 2 0 0 2 2 4 21 11 21 11 22 33 21 12 43 2 2 2 2 0 1 1 2 8 21 11 21 11 22 33 21 12 41 2 2 2 2 0 2 1 1 8 21 11 21 13 23 31 21 33 43 2 2 2 2 0 2 2 0 4 21 11 21 31 42 33 23 13 41 2 2 2 2 1 1 1 1 1 21 11 22 32 51 11 52 33 21 2 2 2 2 2 0 2 0 4 21 11 21 31 21 33 22 13 41 2 2 2 2 2 2 1 2 8

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STAGE 3: GETTING INTO THE SQUARES GROUP
Corners and edges are numbered as in the above diagrams. Moves are coded as follows: 1=L, 2=L2, 3=L', 4=F2, 5=R, 6=R2, 7=R', 8=B2, 9=U2, 10=D2. To accomplish stage 3, proceed as follows: (i) Establish which corners are out of orbit. (ii) Perform moves as indicated in the table below. (iii) Find by calculation the right coset of the squares group in which the permutation of corners now lies. (iv) Refer to the appropriate detailed table. Find the numbers of the positions of the four edge pieces of the FB-slice, i.e. LU, LD, RU, RD. Find the number in the left-hand column corresponding to these, and perform the inverse of the given move. since some cosets are reflections of others, the reader may have to undertake the extra task of reflecting his position to find a suitable table. CORNERS OUT OF ORBIT | INITIAL MOVE 1,5 | - 1,6 | - 1,7 | 4 1,8 | 2 9 1,2,5,6 | 1 1,2,5,7 | 1 4 1,2,5,8 | 3 9 1,2,7,8 | - 1,3,6,8 | 9 1,3,5,7 | 6 4 1,3,5,8 | 1 9 1,2,3.5,6,7 | 1 1,2,3,5,7,8 | 4 5 1,2,3,6,7,8 | 1 5 9 1,2,3,4,5,6,7,8 | 1 5

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page 1 CORNERS OUT OF ORBIT: NONE COSET: G COSET: G(14)(68) 1234 - 1 4 1 4 1 5 8 1 8 7 1235 1 4 1 7 4 10 2 7 1 4 1 4 9 2 9 7 4 7 1236 1 4 1 7 4 10 7 8 1 4 1 4 2 9 1 8 10 7 1237 1 4 1 7 4 10 7 4 1 4 1 4 2 8 9 7 9 7 1238 1 4 1 7 4 9 3 10 1 4 3 4 2 8 10 1 8 3 1245 1 4 1 7 4 10 2 5 1 4 1 4 2 9 1 4 10 7 1246 1 4 1 7 4 10 5 8 1 4 1 4 9 2 9 3 9 3 1247 1 4 1 7 4 10 5 4 1 4 3 4 1 5 8 3 8 10 3 1248 1 4 1 7 4 9 1 9 1 4 1 4 2 8 9 3 8 3 1256 1 4 1 7 4 8 10 5 1 4 1 4 1 5 8 1 4 3 1257 1 4 1 4 8 3 10 7 1 4 1 4 2 9 1 10 5 4 1258 1 4 1 4 8 3 10 5 1 4 3 4 1 5 9 8 1 4 3 1267 1 4 1 4 8 7 8 1 1 4 3 4 2 4 9 1 10 7 1268 1 4 5 9 10 3 4 3 1 4 1 4 2 4 10 3 9 7 1278 1 4 1 7 9 1 9 10 1 4 1 4 2 10 1 8 5 10 1345 1 4 1 7 4 9 3 8 1 4 1 4 10 2 10 7 10 7 1346 1 4 1 4 8 3 9 1 1 4 1 4 1 5 4 5 8 3 1347 1 4 1 7 4 9 3 6 1 4 1 4 2 10 1 4 1 4 1348 1 4 1 7 4 9 2 1 1 4 1 4 2 9 1 8 9 3 1356 1 4 1 9 10 3 9 5 1 4 3 4 2 9 7 4 10 7 1357 1 4 8 1 4 3 7 8 1 4 3 4 2 9 7 9 1 8 1358 1 4 2 9 4 6 10 3 1 4 1 4 2 4 9 7 10 7 1367 1 4 2 9 4 6 9 7 1 4 3 4 2 4 10 1 8 7 1368 1 4 8 1 4 3 7 4 1 4 3 4 2 8 9 1 9 7 1378 1 4 1 7 9 1 9 6 1 4 3 4 2 9 7 9 1 9 1456 1 4 1 9 10 3 9 7 1 4 3 4 2 10 3 9 5 10 1457 1 4 1 7 9 1 8 9 1 4 1 4 2 10 1 8 5 8 1458 1 4 8 1 9 3 7 9 1 4 1 4 2 8 10 3 9 3 1467 1 4 8 1 9 1 5 9 1 4 1 4 1 4 3 10 3 9 7 1468 1 4 1 7 9 1 4 9 1 4 1 4 2 10 1 8 5 4 1478 1 4 1 7 9 1 6 9 1 4 1 4 10 6 9 3 4 7 1567 1 4 8 2 9 8 10 1 1 4 1 4 9 6 10 3 9 7 1568 1 4 8 2 9 4 10 3 1 4 1 4 1 9 3 10 3 8 3 1578 1 4 8 9 4 10 3 4 3 8 3 8 9 6 10 1 9 5 4 1578 1 4 8 9 4 9 7 9 3 8 3 8 3 9 1 10 1 4 1 8 2345 1 4 1 4 8 3 9 3 1 4 1 4 1 5 4 1 9 7 2346 1 4 1 7 4 10 5 9 1 4 1 4 10 2 10 3 4 3 2347 1 4 1 7 4 9 1 6 1 4 1 4 2 9 1 4 9 3 2348 1 4 1 7 4 9 1 4 1 4 1 4 2 10 5 9 5 8 2356 1 4 3 9 10 1 10 5 1 4 1 4 2 9 1 10 5 9 2357 1 4 1 7 9 1 8 10 1 4 1 4 6 9 1 4 5 8 2358 1 4 8 1 9 3 7 10 1 4 1 4 1 4 7 8 7 10 7 2367 1 4 8 1 9 1 5 10 1 4 1 4 2 8 10 3 10 7 2368 1 4 1 7 9 1 4 10 1 4 1 4 2 4 9 3 8 7 2378 1 4 1 7 9 1 6 10 1 4 1 4 10 2 10 3 8 7 2456 1 4 1 9 10 7 4 1 1 4 3 4 2 9 3 9 4 3 2457 1 4 8 1 1 1 5 8 1 4 3 4 2 8 9 5 8 3 2458 1 4 2 9 4 2 10 7 1 4 3 4 6 8 9 1 4 7 2467 1 4 2 9 4 2 9 3 1 4 1 4 2 4 9 3 4 3 2468 1 4 8 1 4 1 5 4 1 4 3 4 6 10 7 10 1 4 2478 1 4 1 7 9 1 9 2 1 4 1 4 1 5 4 1 10 3 2567 1 4 8 2 9 8 10 3 1 4 1 4 1 9 3 10 7 9 7 2568 1 4 8 2 9 4 10 1 1 4 1 4 9 2 9 3 10 7 2578 1 4 8 9 4 10 1 4 3 8 3 8 3 9 1 10 5 9 5 4 2678 1 4 8 9 4 9 5 10 3 8 3 8 9 2 9 1 10 5 8 3456 1 4 1 7 9 1 4 8 1 4 1 4 1 9 3 9 3 9 3 3457 1 4 1 7 9 1 6 8 1 4 3 4 6 4 9 5 10 3 3458 1 4 1 7 9 1 4 2 1 4 1 4 2 8 9 3 4 7 3467 1 4 1 7 9 1 4 6 1 4 1 4 6 4 10 3 8 7 3468 1 4 1 7 9 1 6 4 1 4 3 4 2 8 10 5 9 3 3478 1 4 1 5 9 2 10 1 1 4 1 4 2 10 1 8 5 9 3567 1 4 8 9 4 9 7 8 7 8 7 8 10 6 10 5 9 1 8 3568 1 4 8 9 4 10 3 9 7 8 7 8 7 10 5 9 1 10 1 4 3578 1 4 8 2 9 8 9 5 5 4 5 4 10 6 10 7 9 3 3678 1 4 8 2 9 4 9 7 5 6 5 4 5 10 7 9 3 10 3 4567 1 4 8 9 4 9 5 8 7 8 7 8 7 10 5 9 5 4 5 8 4568 1 4 8 9 4 10 1 10 7 8 7 8 10 2 9 5 10 1 4 4578 1 4 8 2 9 8 9 7 5 4 5 4 5 10 7 9 7 8 7 4678 1 4 8 2 9 4 9 5 5 4 5 4 10 2 9 7 10 3 5678 1 4 1 7 9 4 8 5 1 10 9 8 1 4 2 9 1 9 1

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Page 2 ALL CORNERS IN ORBIT. COSET: G(12) COSET: G(14) 1234 1 4 2 9 1 4 2 4 9 1 1 4 2 9 1 4 2 4 9 1 4 1235 1 4 2 9 1 4 2 10 5 1 4 2 9 3 4 2 9 1 6 4 1236 1 4 2 9 1 4 6 9 7 1 4 2 9 3 4 1 5 8 5 8 1237 1 4 2 9 3 4 6 10 1 4 8 1 4 2 9 3 4 1 5 8 5 4 1238 1 4 2 9 3 4 6 10 3 4 8 1 4 2 9 3 4 2 9 1 6 8 1245 1 4 2 9 1 4 2 10 7 1 4 2 9 3 4 2 9 1 4 10 1246 1 4 2 9 1 4 6 9 5 1 4 2 9 3 4 1 5 8 7 8 1247 1 4 2 9 3 4 2 9 3 4 8 1 4 2 9 3 4 1 5 8 7 4 1248 1 4 2 9 3 4 2 9 1 4 8 1 4 2 9 1 4 2 9 1 10 6 1256 1 4 2 9 1 4 2 4 10 5 1 4 2 9 7 4 2 4 9 3 8 10 1257 1 4 2 9 3 9 1 4 8 5 8 3 1 4 2 9 1 4 2 4 10 5 4 1258 1 4 2 9 3 9 1 9 10 5 8 5 1 4 2 9 7 4 2 4 9 3 10 2 1267 1 4 2 9 3 9 1 4 8 1 9 5 1 4 2 9 7 4 2 4 9 3 10 6 1268 1 4 2 9 3 9 1 4 8 1 9 7 1 4 2 9 1 4 2 4 10 5 8 1278 1 4 2 9 1 4 2 4 10 5 4 8 1 4 2 9 7 4 2 4 9 3 4 10 1345 1 4 2 9 3 4 6 10 3 9 10 1 4 2 9 3 4 2 9 3 9 2 1346 1 4 2 9 3 4 6 10 1 9 10 1 4 2 9 3 4 2 9 1 9 2 1347 1 4 2 9 3 4 6 4 8 9 7 1 4 2 9 3 4 6 10 1 4 10 1348 1 4 2 9 3 4 6 4 8 9 5 1 4 2 9 3 4 6 10 3 8 10 1356 7 4 6 9 5 4 7 4 8 3 10 3 1 4 2 9 1 9 2 10 4 1 9 2 1357 1 4 2 9 3 4 9 6 4 9 5 7 4 6 9 5 4 7 4 8 3 10 3 4 1358 1 4 2 9 3 6 2 4 9 1 6 1 4 2 9 3 4 6 4 9 5 4 1367 1 4 2 9 3 4 1 5 9 4 3 1 4 2 9 3 4 6 4 10 1 4 1368 1 4 2 9 3 4 9 6 8 9 7 5 8 6 9 7 8 5 8 4 1 10 1 8 1378 1 4 2 9 3 4 1 4 8 5 10 5 1 4 2 9 1 9 2 9 8 5 9 6 1456 7 4 6 9 5 4 7 9 10 7 8 1 1 4 2 9 3 4 2 4 9 1 4 1457 1 4 2 9 3 4 1 5 10 4 3 1 4 2 9 1 4 2 1 10 5 4 10 1458 1 4 2 9 3 4 9 2 8 9 3 1 4 2 9 7 4 9 2 4 9 3 4 1467 1 4 2 9 3 4 9 2 4 9 1 1 4 2 9 7 4 9 2 8 9 1 4 1468 1 4 2 9 3 4 1 5 10 8 7 1 4 2 9 7 4 2 4 9 1 9 2 1478 1 4 2 9 3 4 1 4 8 1 8 3 1 4 2 9 3 4 2 4 9 1 8 1567 1 4 2 9 3 4 1 5 4 2 1 1 4 2 9 3 4 1 5 4 3 4 1568 1 4 2 9 3 4 1 5 4 3 6 1 4 2 9 3 4 2 10 2 7 8 1578 1 4 2 9 3 4 6 9 5 4 8 1 4 2 9 3 4 2 10 2 7 4 1678 1 4 2 9 3 4 6 9 7 4 8 1 4 2 9 3 4 1 5 4 3 8 2345 1 4 2 9 3 4 2 9 1 9 10 1 4 2 9 3 4 2 9 1 4 9 2346 1 4 2 9 3 4 2 9 3 9 10 1 4 2 9 3 4 2 9 3 8 9 2347 1 4 2 9 3 4 2 4 8 10 5 1 4 2 9 3 4 2 9 3 4 9 2348 1 4 2 9 3 4 2 4 8 10 7 1 4 2 9 3 4 2 9 1 8 9 2356 5 4 6 10 7 4 5 10 9 5 8 3 1 4 2 9 3 4 2 4 10 5 4 2357 1 4 2 9 3 4 1 5 9 4 1 1 4 2 9 1 4 2 4 10 5 4 9 2358 1 4 2 9 3 4 9 2 4 10 5 1 4 2 9 7 4 9 2 8 10 5 4 2367 1 4 2 9 3 4 9 2 8 10 7 1 4 1 9 7 4 9 2 4 10 7 4 2368 1 4 2 9 3 4 1 5 9 8 5 1 4 2 9 1 9 2 9 8 5 4 10 2378 1 4 2 9 3 4 1 9 10 1 8 7 1 4 2 9 3 4 2 4 10 5 8 2456 5 4 6 10 7 4 5 4 8 1 9 1 1 4 2 9 1 9 2 10 4 1 9 6 2457 1 4 2 9 3 4 9 2 4 9 3 5 4 6 10 7 4 5 4 8 1 9 1 4 2458 1 4 2 9 3 4 2 8 9 2 3 1 4 2 9 3 4 2 4 10 7 4 2467 1 4 2 9 3 1 1 5 9 8 7 1 4 2 9 3 4 2 4 9 3 4 2458 1 4 2 9 3 4 9 2 8 9 1 7 8 6 10 5 8 7 8 4 3 9 3 8 2478 1 4 2 9 3 4 1 4 8 1 8 1 1 4 2 9 1 4 2 8 9 1 9 2 2567 1 4 2 9 3 4 1 5 4 2 3 1 4 2 9 3 4 1 5 4 1 4 2568 1 4 2 9 3 4 1 5 4 1 6 1 4 2 9 3 4 2 10 5 9 6 2578 1 4 2 9 3 4 2 10 7 4 8 1 4 2 9 3 4 2 10 7 4 9 2678 1 4 2 9 3 4 2 10 5 4 8 1 4 2 9 3 4 1 5 4 1 8 3456 1 4 2 9 1 4 2 4 10 5 9 10 1 4 2 9 7 4 2 4 9 3 8 9 3457 7 4 6 9 5 9 7 4 8 7 9 1 1 4 2 9 1 4 1 5 10 8 5 8 3458 5 8 6 9 7 9 5 8 4 5 9 1 7 4 6 9 1 4 6 4 9 5 10 6 3467 7 4 6 9 5 9 7 4 8 7 9 3 5 8 6 9 3 8 6 8 9 7 10 6 3468 5 8 6 9 7 9 5 8 4 5 9 3 1 4 2 9 1 4 1 5 10 8 5 4 3478 1 4 2 9 3 6 4 2 4 10 5 1 4 2 9 7 4 2 4 9 3 4 9 3567 1 4 2 9 3 4 6 9 7 9 10 1 4 2 9 3 4 2 10 7 10 6 3568 1 4 2 9 3 4 6 9 5 9 10 1 4 2 9 3 4 2 10 5 10 6 3578 1 4 2 9 3 4 6 4 8 10 3 1 4 2 9 3 4 6 9 5 4 9 3678 1 4 2 9 3 4 6 4 8 10 1 1 4 2 9 3 4 6 9 7 8 9 4567 1 4 2 9 3 4 2 10 5 9 10 1 4 2 9 3 4 2 10 5 4 10 4568 1 4 2 9 3 4 2 10 7 9 10 1 4 2 9 3 4 2 10 7 8 10 4578 1 4 2 9 3 4 2 4 8 9 1 1 4 2 9 3 4 2 10 7 4 10 4678 1 4 2 9 3 4 2 4 8 9 3 1 4 2 9 3 4 2 10 5 8 10 5678 1 4 2 9 3 6 4 2 4 9 1 1 4 2 9 1 8 2 4 9 5 4

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Page 3 CORNERS:(18)(27) COSET:G COSET: G(14)(68) 1234 1 4 1 4 9 4 3 4 1 1 4 3 8 10 2 9 3 4 1 1235 1 4 1 4 10 8 3 9 3 3 9 3 9 3 9 5 10 3 4 3 1236 1 4 1 4 10 8 3 9 1 3 9 3 9 3 9 5 10 3 4 1 1237 1 4 1 4 8 1 9 4 5 1 4 3 8 2 8 10 3 4 1 1238 1 4 1 6 9 10 3 8 1 1 4 3 9 2 8 5 9 8 5 1245 1 4 2 9 2 9 4 9 1 3 9 3 9 3 9 1 8 7 8 1 1246 1 4 1 4 10 8 7 4 7 3 9 3 9 3 9 1 8 7 8 3 1247 1 4 1 6 9 10 7 9 7 1 4 3 9 2 8 1 4 9 1 1248 1 4 1 4 8 1 9 4 7 1 4 3 8 2 8 10 7 10 7 1256 1 4 9 3 4 8 3 10 8 1 1 4 3 4 2 4 10 3 9 5 1257 1 4 1 4 9 8 3 10 7 1 4 3 4 9 6 10 3 10 7 1258 1 4 1 4 9 8 3 10 5 1 4 3 4 9 2 9 7 4 1 1267 1 4 1 4 9 8 7 8 1 1 4 3 4 9 2 9 3 9 5 1268 1 4 1 10 8 9 7 4 7 1 4 3 4 9 2 9 3 9 7 1278 7 10 7 9 4 9 5 9 7 9 10 1 4 9 1 9 2 4 3 9 5 1345 1 4 1 6 4 8 3 4 1 1 4 3 4 2 4 9 3 4 1 1346 1 4 1 8 9 4 3 9 1 1 4 3 10 2 9 4 3 4 1 1347 1 4 1 5 4 9 4 10 7 1 4 3 4 2 8 9 7 9 7 1348 1 4 1 5 4 9 4 10 5 1 4 3 4 2 8 9 3 8 1 1356 1 4 1 4 9 8 3 10 1 1 4 3 6 9 2 4 7 8 1 1357 1 4 1 7 4 2 4 1 6 1 4 3 8 2 8 9 3 9 5 1358 1 4 1 5 4 2 1 4 8 1 4 3 4 1 5 4 1 10 7 1367 1 4 1 5 4 1 4 8 2 1 4 3 4 2 10 1 9 8 7 1368 1 4 1 7 4 2 4 2 7 1 4 3 4 10 2 10 3 8 3 1378 1 4 1 4 8 1 4 2 5 1 4 3 9 2 8 5 10 8 5 1456 1 4 1 4 9 8 7 8 7 1 4 3 6 9 6 8 3 9 7 1457 1 4 1 5 4 2 7 9 10 1 4 3 4 1 5 4 1 10 5 1458 1 4 1 7 4 2 4 2 3 1 4 3 8 2 8 9 3 9 7 1467 1 4 1 7 4 1 5 9 3 1 4 3 4 10 6 9 3 4 1 1468 1 4 1 5 4 2 3 9 10 1 4 3 4 2 10 1 9 8 5 1478 1 4 1 4 8 1 4 2 7 1 4 3 9 2 8 1 8 9 3 1567 1 4 1 4 10 8 3 9 7 1 4 9 1 10 6 8 3 4 1 1568 1 4 1 4 10 8 3 9 5 1 4 9 1 10 2 4 3 8 3 1578 1 4 1 8 9 4 3 9 5 1 4 3 8 2 10 1 8 9 3 1678 1 4 1 6 4 8 7 10 1 1 4 1 4 10 2 10 1 9 5 2345 1 4 1 8 9 4 3 9 3 1 4 3 10 2 9 4 3 4 3 2346 1 4 1 6 4 8 3 4 3 1 4 3 4 2 4 9 3 4 3 2347 1 4 2 4 9 4 9 4 1 1 4 3 4 6 4 10 3 4 1 2348 1 4 2 9 2 9 8 9 7 1 4 3 4 2 8 9 3 8 3 2356 1 4 1 4 9 8 3 10 3 1 4 3 6 9 2 4 3 10 7 2357 1 4 1 5 4 2 3 4 8 1 4 3 4 6 9 1 9 4 5 2358 1 4 1 7 4 2 9 4 7 1 4 3 4 10 2 10 3 8 1 2367 1 4 1 7 4 1 5 9 5 1 4 3 8 6 4 10 3 10 7 2368 1 4 1 5 4 2 4 8 3 1 4 3 4 2 10 1 8 2 3 2378 1 4 1 4 8 1 10 4 5 1 4 3 9 6 4 1 4 10 3 2456 1 4 1 10 8 9 7 4 1 1 4 3 6 9 2 4 3 10 5 2457 1 4 1 7 4 1 5 9 1 1 4 3 4 10 2 10 7 9 7 2458 1 4 1 5 4 1 4 8 6 1 4 3 4 2 10 5 4 10 3 2467 1 4 1 5 4 1 6 4 8 1 4 3 4 1 5 4 5 4 3 2458 1 4 1 7 4 1 5 9 7 1 4 3 8 2 8 9 7 4 1 2478 1 4 1 4 8 1 10 4 7 1 4 3 9 2 8 1 8 9 1 2567 1 4 1 4 10 8 7 4 1 1 4 9 1 10 2 4 7 9 7 2568 1 4 2 9 2 9 4 9 7 1 4 9 1 10 2 4 3 8 1 2578 1 4 1 6 4 8 3 4 5 1 4 3 4 1 5 8 1 8 3 2678 1 4 1 8 9 4 7 4 1 1 4 3 8 2 10 1 8 9 1 3456 1 10 1 9 4 9 3 9 1 9 10 1 4 9 1 10 2 8 3 9 5 3457 1 4 1 4 8 1 4 2 3 1 4 3 4 2 9 1 4 9 3 3458 1 4 1 4 8 1 4 1 6 1 4 3 4 2 9 5 9 8 5 3467 1 4 1 4 8 1 4 2 1 1 4 3 4 2 9 1 4 9 1 3468 1 4 1 4 8 1 4 3 6 1 4 3 4 6 10 1 8 10 3 3478 1 4 1 8 9 8 3 8 1 1 4 5 8 10 2 10 1 10 5 3567 1 4 1 6 9 10 3 8 7 1 4 3 4 2 8 10 3 10 7 3568 1 4 1 4 8 5 8 9 5 1 4 3 10 2 4 5 9 8 5 3578 1 4 1 5 4 9 4 10 3 3 9 3 9 4 6 8 3 4 10 1 3678 1 4 1 5 4 9 4 10 1 3 9 3 9 4 6 8 3 4 10 3 4567 1 4 1 4 8 1 9 4 1 1 4 3 10 2 4 1 4 9 1 4568 1 4 1 6 9 10 3 8 5 1 4 3 4 2 8 10 3 10 5 4578 1 4 2 9 2 9 8 9 1 3 9 3 9 4 2 4 3 8 9 3 4678 1 4 2 4 9 4 9 4 7 3 9 3 9 4 2 4 3 8 9 1 5678 1 4 1 4 9 4 3 4 5 1 4 5 4 2 8 9 1 4 1

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CORNERS: (13). COSET:G(12) COSET: G(14) Page 5 1234 1 4 3 4 1 4 9 1 9 1 9 1 4 1 4 2 4 3 4 5 9 5 1235 1 4 3 6 9 7 10 1 8 10 1 1 4 1 4 1 4 2 9 10 4 2 1236 1 4 3 4 1 10 3 10 3 10 8 1 4 1 4 1 4 2 3 1 4 10 1237 1 4 3 4 5 10 4 5 4 3 10 1 4 1 4 1 4 2 8 9 10 2 1238 1 4 3 4 1 8 10 1 8 1 10 1 4 1 4 1 4 2 8 6 9 10 1245 1 4 3 6 9 3 4 5 4 9 3 1 4 1 4 1 4 2 4 9 2 9 1246 1 4 3 6 9 8 1 4 3 4 3 1 4 1 4 1 4 2 4 2 9 2 1247 1 4 3 6 9 4 5 8 3 4 3 1 4 1 4 1 4 8 9 10 2 8 1248 1 4 3 4 9 7 4 9 7 9 7 1 4 1 4 1 4 2 8 2 6 9 1256 1 4 3 4 1 4 9 2 3 10 5 1 4 1 4 1 5 4 8 2 3 4 1257 1 4 3 4 5 8 3 8 10 3 10 1 4 1 4 2 8 7 4 9 4 8 1258 1 4 3 4 8 9 1 9 7 8 1 1 4 1 4 9 2 9 6 4 5 8 1267 1 4 3 6 10 8 5 9 3 8 3 1 4 1 4 1 7 4 8 2 1 4 1268 1 4 3 4 5 8 3 4 9 5 9 1 4 1 4 2 4 7 9 1 4 1 1278 1 4 3 6 10 3 4 9 3 8 1 10 1 4 1 4 2 4 9 4 10 5 8 1345 3 8 1 6 8 1 9 8 5 4 3 1 4 1 4 1 4 1 6 8 1 8 1346 1 4 3 6 4 3 9 4 7 8 1 1 4 1 4 1 4 1 5 9 1 5 1347 1 4 3 4 1 8 10 1 8 1 6 1 4 1 4 1 4 2 6 9 6 8 1348 1 4 3 4 1 8 10 1 8 2 3 1 4 1 4 1 4 2 8 9 6 9 1356 1 4 3 4 5 9 3 4 8 10 5 1 4 1 4 1 7 8 3 10 6 8 1357 1 4 3 4 1 8 3 4 8 9 5 1 4 1 4 2 4 10 2 4 5 8 1358 1 4 3 4 1 9 1 6 10 4 5 1 4 1 4 2 8 7 4 2 4 8 1367 1 4 3 6 8 3 9 5 8 9 3 1 4 1 4 2 8 10 4 7 8 2 1368 1 4 3 6 4 3 4 5 4 10 3 1 4 1 4 1 5 9 2 4 7 8 1378 1 4 3 6 10 3 4 9 7 9 7 1 4 1 4 10 7 4 10 8 6 4 1456 1 4 3 4 1 4 3 9 10 8 3 1 4 1 4 1 7 8 3 6 8 10 1457 1 4 3 4 1 9 1 6 10 4 7 1 4 1 4 2 4 8 9 8 1 4 1458 1 4 3 4 1 8 3 4 8 9 7 1 4 1 4 1 5 10 1 4 2 4 1467 1 4 3 6 4 7 10 1 8 9 1 1 4 1 4 2 4 8 5 8 9 2 1468 1 4 3 6 8 7 4 1 4 10 1 1 4 1 4 2 4 9 10 4 1 8 1478 1 4 3 6 10 3 4 9 3 8 1 1 4 1 4 2 8 7 10 4 9 4 1567 5 8 7 8 5 4 10 5 8 3 1 4 1 4 6 4 10 4 10 5 8 1568 7 4 5 4 7 8 10 7 4 1 1 4 1 4 9 4 1 5 9 3 4 1578 1 4 3 4 1 8 10 1 4 7 9 1 4 1 4 8 10 8 2 10 5 8 1678 1 4 3 4 5 10 4 5 8 5 9 1 4 1 4 10 5 8 9 3 10 5 2345 1 4 3 4 1 4 1 4 2 8 1 1 4 1 4 1 4 6 10 8 10 4 2346 1 4 3 4 1 4 1 8 2 8 3 1 4 1 4 1 4 2 6 8 2 8 2347 1 4 3 4 2 4 1 4 1 8 1 1 4 1 4 1 4 9 4 9 6 4 2348 1 4 3 4 1 10 3 4 9 7 9 1 4 1 4 1 4 6 8 10 4 8 2356 1 4 3 4 1 4 3 4 5 4 8 1 4 1 4 2 8 7 9 4 9 8 2357 1 4 3 4 1 8 1 10 6 10 5 1 4 1 4 1 5 10 1 10 6 8 2358 1 4 3 4 1 8 2 1 9 2 7 1 4 1 4 1 5 10 1 8 6 8 2367 1 4 3 4 1 8 9 5 4 5 9 1 4 1 4 1 4 1 9 2 5 4 2368 1 4 3 4 1 8 1 9 2 9 7 1 4 1 4 1 5 10 1 9 6 4 2378 1 4 3 4 1 4 3 4 7 9 6 1 4 1 4 1 7 8 2 3 8 9 2456 1 4 3 4 1 9 8 1 10 8 5 1 4 1 4 1 9 2 1 8 1 8 2457 1 4 3 4 1 8 1 6 9 3 9 1 4 1 4 1 5 4 10 2 3 8 2458 1 4 3 4 1 8 1 10 6 10 7 1 4 1 4 1 4 6 4 1 10 5 2467 1 4 3 4 1 8 1 9 2 9 5 1 4 1 4 2 4 10 8 7 4 6 2458 1 4 3 4 10 3 4 9 7 4 1 1 4 1 4 1 8 2 8 1 10 5 2478 1 4 3 1 1 4 2 1 4 7 9 1 4 1 4 1 7 8 3 9 6 4 2567 1 4 3 4 1 4 2 10 7 4 1 1 4 1 4 1 6 8 9 1 10 5 2568 1 4 3 4 1 4 6 9 3 10 7 1 4 1 4 1 5 9 2 8 3 4 2578 3 8 1 8 2 8 3 4 3 10 7 1 4 1 4 1 9 2 3 10 7 4 2678 1 4 3 4 2 4 1 8 1 10 5 1 4 1 4 2 4 9 2 9 5 6 3456 1 4 3 4 1 9 3 4 1 9 4 1 4 1 4 1 7 10 4 2 5 4 3457 1 4 3 4 1 8 3 9 5 9 2 1 4 1 4 1 5 10 1 6 8 10 3458 1 4 3 4 1 8 1 6 10 5 10 1 4 1 4 1 5 10 2 8 7 8 3467 1 4 3 4 1 8 1 6 9 1 9 1 4 1 4 3 8 3 9 1 9 6 3468 1 4 3 4 1 8 3 10 1 10 2 1 4 1 4 1 4 2 4 5 9 5 3478 1 4 3 4 1 4 1 6 8 5 9 1 4 1 4 1 6 4 10 1 8 1 3567 1 4 3 4 9 3 10 8 3 9 7 1 4 1 4 3 8 10 1 10 7 10 3568 1 4 3 4 1 8 9 1 9 2 7 1 4 1 4 8 2 10 7 10 6 4 3578 1 4 3 4 10 3 8 5 10 3 9 1 4 1 4 9 2 9 6 8 1 4 3678 1 4 3 6 9 3 4 1 9 4 7 1 4 1 4 1 6 4 1 4 8 7 4567 1 4 3 4 1 4 3 4 9 10 3 1 4 1 4 2 4 9 2 4 1 4 4568 1 4 3 4 9 3 10 8 3 9 5 1 4 1 4 1 4 1 6 4 5 4 4578 1 4 3 6 10 3 4 1 9 3 9 1 4 1 4 2 8 9 10 3 4 9 4678 1 4 3 6 9 3 4 1 9 4 5 1 4 1 4 1 6 9 7 4 10 1 5678 1 4 3 6 4 9 1 10 3 4 1 1 4 1 4 3 8 9 5 10 3 9

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CORNERS: (15). COSET: G(24). page 7 1234 1 4 3 9 1 7 10 3 4 5 1235 1 4 3 9 1 4 1 6 4 8 1236 1 4 3 9 2 9 1 10 4 3 1237 1 4 3 10 1 9 8 5 9 10 1238 1 4 3 9 10 6 4 3 10 1 1245 1 4 3 9 1 4 1 4 8 2 1246 1 4 3 9 1 9 3 4 10 8 1247 1 4 3 9 8 2 10 3 9 3 1248 1 4 1 4 3 10 8 5 10 2 1256 1 4 3 9 4 10 8 3 10 1 1257 1 4 3 10 1 8 7 9 8 10 1258 1 4 1 4 2 1 10 5 4 8 1267 1 4 3 9 1 10 5 9 4 2 1268 1 4 3 10 1 8 7 10 4 9 1278 1 4 3 9 1 8 9 3 10 2 1345 1 4 3 9 1 8 1 7 8 1 1346 1 4 3 9 3 10 2 8 10 7 1347 1 4 3 10 1 4 10 4 10 1 1348 1 4 3 10 8 3 8 3 9 6 1356 1 4 3 9 2 10 5 9 4 3 1357 1 4 3 9 5 4 3 9 4 9 1358 1 4 3 10 1 8 5 10 4 2 1367 1 4 3 9 2 8 9 3 10 3 1368 1 4 3 9 5 4 3 10 8 10 1378 1 4 3 9 5 10 8 5 9 10 1456 1 4 3 9 3 10 4 2 4 7 1457 1 4 3 10 1 6 10 6 9 1 1458 1 4 3 10 1 10 6 9 10 1 1467 1 4 3 10 4 9 10 3 8 5 1468 1 4 3 10 3 4 10 6 9 3 1478 1 4 3 10 4 6 10 3 4 7 1567 1 4 3 9 4 1 8 6 9 3 1568 1 4 3 9 4 2 1 10 8 3 1578 3 8 9 3 10 8 7 4 10 7 1678 1 4 9 1 10 4 5 8 10 5 2345 1 4 3 10 1 10 8 5 4 8 2346 1 4 3 9 1 9 1 9 4 6 2347 1 4 3 9 1 4 2 1 9 10 2348 1 4 3 9 1 4 1 9 10 6 2356 1 4 3 9 1 4 2 9 10 5 2357 1 4 3 9 10 8 9 3 10 1 2358 1 4 3 9 10 2 4 7 9 1 2367 1 4 3 9 6 8 9 3 8 7 2368 1 4 3 10 1 9 3 6 4 1 2378 1 4 3 4 10 2 4 3 8 7 2456 1 4 3 9 1 4 3 7 8 3 2457 1 4 3 9 1 8 1 4 1 5 2458 1 4 3 9 1 8 7 9 2 10 2467 1 4 3 9 1 8 7 10 6 9 2458 1 4 3 9 1 5 8 1 4 5 2478 1 4 3 9 4 9 10 7 10 1 2567 1 4 3 4 2 8 5 10 8 3 2568 1 4 3 10 1 8 10 4 9 1 2578 1 4 3 10 4 8 2 3 4 7 2678 1 4 1 4 9 1 10 5 9 6 3456 1 4 3 9 4 3 10 5 9 10 3457 1 4 3 9 1 6 4 6 8 5 3458 1 4 3 9 1 4 6 4 8 5 3467 1 4 3 9 1 9 8 9 4 5 3468 1 4 3 9 1 8 9 6 10 5 3478 1 4 3 2 8 7 9 1 3567 1 4 3 9 1 8 9 4 8 5 3568 1 4 3 9 1 10 2 9 4 5 3578 1 4 3 10 1 9 8 9 10 1 3678 1 4 3 9 8 9 8 3 4 5 4567 1 4 3 9 3 8 6 8 10 7 4568 1 4 3 9 1 5 4 5 8 5 4578 1 4 3 9 4 9 4 7 9 1 4678 1 4 3 9 8 2 9 3 4 7 5678 1 9 10 8 3 9 4 3 9 1

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MOVES IN THE SQUARES GROUP WHICH FIX CORNERS. KEY: 1=L; 2=F; 3=R; 4=B; 5=U; 6=D (ALL 180 DEGREE TURNS). THE NUMBERS ON THE RIGHT GIVE THE PERMUTATION OF EACH SLICE. THE ORDER IN WHICH SLICES ARE CONSIDERED IS FB,UD,LR. FB SLICE: 1=LU, 2=LD, 3=RD, 4=RU. UD SLICE: 1=LB, 2=LF, 3=RF, 4=RB. LR SLICE: 1=FU, 2=FD, 3=BD, 4=BU. THE NUMBER ON EXTREME RIGHT IS THE QUANTITY OF SYMMETRICALLY EQUIVALENT POSITIONS PAGE 1: AT LEAST ONE SLICE FIXED. FB UD LR 1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 3 4 3 (12) (34) 12 1 2 1 2 1 2 1 2 1 4 3 4 (12)(34) 6 1 2 1 2 1 2 1 2 3 2 1 4 (12)(34) (12)(34) 3 1 2 1 2 1 2 1 3 5 1 3 6 (12) (1324) 24 1 2 1 2 1 3 2 5 1 4 1 5 (12)(34) (123) 48 1 2 1 2 1 3 2 5 2 3 2 5 (1234) (12) 24 1 2 1 2 1 3 4 5 1 4 3 6 (123) 24 1 2 1 2 1 3 4 5 2 1 4 6 (24) (12) 24 1 2 1 2 1 5 1 2 5 2 1 5 (23) (12) 6 1 2 1 2 1 5 1 2 6 4 3 6 (14) (12) 6 1 2 1 2 1 5 2 1 2 5 2 1 (14) (12) 24 1 2 1 2 1 5 2 1 4 6 4 1 (1243) (12) 24 1 2 1 2 3 2 5 1 3 2 4 5 (1324) (1423) 12 1 2 1 2 3 5 1 4 5 4 3 6 (23) (1423) 24 1 2 1 2 4 1 2 1 5 1 3 6 (12)(34) (13)(24) 6 1 2 1 2 4 1 2 5 2 4 5 1 (14)(23) (14)(23) 6 1 2 1 2 4 1 4 1 5 1 3 5 (13)(24) (13)(24) 3 1 2 1 2 4 1 5 1 3 5 4 3 (13)(24) (12)(34) 6 1 2 1 2 4 3 2 5 6 3 5 6 (12)(34) (12)(34) 6 1 2 1 2 4 3 5 2 5 4 6 1 (1234) (12) 24 1 2 1 2 4 5 1 2 1 2 4 5 (13)(24) (134) 48 1 2 1 2 4 5 1 3 2 3 2 6 (1432) (1423) 24 1 2 1 2 4 5 1 4 1 2 4 6 (14)(23) (142) 48 1 2 1 2 4 5 2 5 3 4 6 1 (1234) (12) 24 1 2 1 2 5 1 3 6 4 3 2 1 (13)(24) 3 1 2 1 2 5 1 4 3 2 4 5 1 (1324) (24) 24 1 2 1 2 5 1 5 2 1 5 2 1 (13) (24) 12 1 2 1 2 5 1 5 2 5 2 1 5 (123) (142) 48 1 2 1 2 5 1 5 2 6 4 3 6 (134) (142) 48 1 2 1 2 5 2 1 5 1 2 5 2 (12) (13) 24 1 2 1 2 5 2 1 5 1 4 6 4 (1324) (13) 24 1 2 1 2 5 2 5 4 6 3 4 1 (1243) (12) 24 1 2 1 3 5 1 2 1 2 4 6 2 (1423) (1342) 24 1 2 1 3 5 1 2 5 2 3 5 3 (1324) (1432) 24 1 2 1 3 5 1 3 2 5 2 5 3 (1432) (1432) 12 1 2 1 5 2 5 1 5 1 2 5 2 (142) (123) 48 1 2 1 5 2 5 1 5 1 4 6 4 (243) (123) 48 2 1 2 1 3 5 1 2 5 2 3 5 3 2 (1324) (5867) 6 4 1 2 1 3 5 1 2 5 2 3 5 3 4 (1324) (5768) 6 End of execution.

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MOVES IN THE SQUARES GROUP WHICH FIX CORNERS. KEY: 1=L; 2=F; 3=R; 4=B; 5=U; 6=D (ALL 180 DEGREE TURNS). THE NUMBERS ON THE RIGHT GIVE THE PERMUTATION OF EACH SLICE. THE ORDER IN WHICH SLICES ARE CONSIDERED IS FB,UD,LR. FB SLICE: 1=LU, 2=LD, 3=RD, 4=RU. UD SLICE: 1=LB, 2=LF, 3=RF, 4=RB. LR SLICE: 1=FU, 2=FD, 3=BD, 4=BU. THE NUMBER ON EXTREME RIGHT IS THE QUANTITY OF SYMMETRICALLY EQUIVALENT POSITIONS PAGE 2: 3 3-CYCLES FB UD LR 1 2 1 2 5 1 5 1 2 5 2 5 (124) (123) (142) 16 1 2 1 2 5 1 5 1 2 6 4 6 (124) (134) (142) 48 1 2 1 2 5 1 5 1 4 5 2 6 (132) (134) (142) 16 1 2 1 2 5 1 5 1 4 6 4 5 (132) (123) (142) 48 1 2 1 5 2 5 2 1 5 1 5 2 (142) (123) (142) 48 1 2 1 5 2 5 2 1 5 3 5 4 (123) (243) (134) 48 1 2 1 5 2 5 2 1 6 1 6 4 (123) (134) (134) 48 1 2 1 5 2 5 2 1 6 3 6 2 (142) (142) (142) 48 1 2 1 5 2 5 2 3 5 1 6 2 (142) (142) (123) 96 1 2 1 5 2 5 2 3 5 3 6 4 (123) (134) (243) 96

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MOVES IN THE SQUARES GROUP WHICH FIX CORNERS. KEY: 1=L; 2=F; 3=R; 4=B; 5=U; 6=D (ALL 180 DEGREE TURNS). THE NUMBERS ON THE RIGHT GIVE THE PERMUTATION OF EACH SLICE. THE ORDER IN WHICH SLICES ARE CONSIDERED IS FB,UD,LR. FB SLICE: 1=LU, 2=LD, 3=RD, 4=RU. UD SLICE: 1=LB, 2=LF, 3=RF, 4=RB. LR SLICE: 1=FU, 2=FD, 3=BD, 4=BU. THE NUMBER ON EXTREME RIGHT IS THE QUANTITY OF SYMMETRICALLY EQUIVALENT POSITIONS PAGE 3: NO SLICE FIXED; 2 3-CYCLES AND ONE DOUBLE TRANSPOSITION. FB UD LR 1 2 1 2 5 1 5 2 5 4 1 6 (14)(23) (142) (134) 96 1 2 1 2 5 1 5 2 6 2 3 5 (14)(23) (243) (134) 96 1 2 1 2 5 1 5 4 5 4 3 5 (13)(24) (243) (134) 48 1 2 1 2 5 1 5 4 6 2 1 6 (13)(24) (142) (134) 48 1 2 1 5 2 5 1 5 3 2 6 2 (134) (243) (13)(24) 48 1 2 1 5 2 5 1 5 3 4 5 4 (123) (243) (13)(24) 48 1 2 1 5 2 5 2 1 2 6 3 6 (142) (12)(34) (142) 96 1 2 1 5 2 5 2 1 4 5 3 5 (123) (14)(23) (134) 96 End of execution.

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MOVES IN THE SQUARES GROUP WHICH FIX CORNERS. KEY: 1=L; 2=F; 3=R; 4=B; 5=U; 6=D (ALL 180 DEGREE TURNS). THE NUMBERS ON THE RIGHT GIVE THE PERMUTATION OF EACH SLICE. THE ORDER IN WHICH SLICES ARE CONSIDERED IS FB,UD,LR. FB SLICE: 1=LU, 2=LD, 3=RD, 4=RU. UD SLICE: 1=LB, 2=LF, 3=RF, 4=RB. LR SLICE: 1=FU, 2=FD, 3=BD, 4=BU. THE NUMBER ON EXTREME RIGHT IS THE QUANTITY OF SYMMETRICALLY EQUIVALENT POSITIONS PAGE 4: NO SLICE FIXED; AT LEAST TWO DOUBLE TRANSPOSITIONS FB UD LR 1 2 1 2 1 3 2 1 2 1 5 6 (13)(24) (13)(24) (14)(23) 6 1 2 1 2 1 3 2 1 2 5 6 3 (13)(24) (14)(23) (14)(23) 6 1 2 1 2 1 3 2 1 5 6 4 3 (13)(24) (12)(34) (14)(23) 3 1 2 1 2 1 3 2 3 4 3 5 6 (13)(24) (13)(24) (13)(24) 1 1 2 1 2 1 3 2 5 1 2 1 6 (13)(24) (13)(24) (243) 24 1 2 1 2 1 3 4 3 4 1 5 6 (14)(23) (13)(24) (14)(23) 3 1 2 1 2 1 3 4 3 4 5 6 3 (14)(23) (14)(23) (14)(23) 6 1 2 1 2 1 3 4 5 1 2 3 5 (14)(23) (13)(24) (243) 48 1 2 1 2 1 3 5 2 1 2 6 3 (234) (14)(23) (14)(23) 48 1 2 1 2 1 3 5 2 3 4 5 1 (234) (14)(23) (13)(24) 48 1 2 1 2 4 1 5 2 1 2 6 2 (243) (12)(34) (14)(23) 24 1 2 1 2 4 5 1 2 5 6 1 5 (12)(34) (12)(34) (142) 24 1 2 1 2 5 2 4 5 4 3 2 1 (12)(34) (14)(23) (14)(23) 2 1 2 3 4 5 2 1 2 1 3 5 1 (234) (14)(23) (12)(34) 24 End of execution.

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MOVES IN THE SQUARES GROUP WHICH FIX CORNERS. KEY: 1=L; 2=F; 3=R; 4=B; 5=U; 6=D (ALL 180 DEGREE TURNS). THE NUMBERS ON THE RIGHT GIVE THE PERMUTATION OF EACH SLICE. THE ORDER IN WHICH SLICES ARE CONSIDERED IS FB,UD,LR. FB SLICE: 1=LU, 2=LD, 3=RD, 4=RU. UD SLICE: 1=LB, 2=LF, 3=RF, 4=RB. LR SLICE: 1=FU, 2=FD, 3=BD, 4=BU. THE NUMBER ON EXTREME RIGHT IS THE QUANTITY OF SYMMETRICALLY EQUIVALENT POSITIONS PAGE 5: NO SLICE FIXED; TWO 2-CYCLES. FB UD LR 1 2 1 2 1 2 1 5 1 5 1 5 (12) (12) (124) 48 1 2 1 2 1 2 1 5 1 6 3 6 (12) (34) (124) 48 1 2 1 2 1 2 1 5 6 3 5 6 (12) (12)(34) (12) 24 1 2 1 2 1 2 5 1 3 2 4 6 (34) (13)(24) (34) 12 1 2 1 2 1 3 5 1 5 3 6 2 (12) (13) (123) 96 1 2 1 2 1 3 5 1 6 1 5 2 (12) (24) (123) 96 1 2 1 2 1 3 5 2 5 3 2 5 (124) (34) (12) 96 1 2 1 2 1 3 5 2 6 1 4 6 (124) (12) (12) 96 1 2 1 2 1 5 1 4 5 2 3 6 (12)(34) (14) (12) 12 1 2 1 2 1 5 1 4 6 4 1 5 (12)(34) (23) (12) 12 1 2 1 2 1 5 1 5 2 5 1 5 (12) (132) (12) 24 1 2 1 2 1 5 1 5 2 6 3 6 (12) (143) (12) 24 1 2 1 2 1 5 1 6 4 5 3 6 (12) (124) (12) 48 1 2 1 2 4 3 5 1 5 4 1 5 (12)(34) (14) (14) 24 1 2 1 2 4 5 1 2 5 3 6 1 (12)(34) (12) (24) 24 1 2 1 2 4 5 1 3 2 1 2 5 (13) (13)(24) (34) 24 1 2 1 2 4 5 1 5 3 6 4 1 (12)(34) (14) (24) 24 1 2 1 2 5 1 2 1 2 4 5 3 (34) (14)(23) (13) 24 1 2 1 2 5 1 5 4 3 6 4 1 (12)(34) (13) (24) 24 1 2 1 2 5 2 3 5 1 4 6 2 (12) (24) (12)(34) 24 1 2 1 3 2 5 2 1 5 1 4 6 (13)(24) (34) (34) 24 1 2 1 3 5 1 2 3 2 4 6 4 (12) (12)(34) (14) 24 1 2 1 3 5 1 2 5 4 1 5 1 (34) (24) (13)(24) 24 1 2 1 3 5 1 3 2 5 4 5 1 (13) (24) (13)(24) 12 1 2 1 3 5 1 5 1 2 1 4 6 (24) (23) (234) 96 1 2 1 3 5 1 5 3 2 1 2 6 (13) (23) (132) 96 1 2 1 5 1 5 1 2 5 2 5 1 (24) (132) (24) 48 1 2 1 5 1 5 1 2 6 4 6 1 (24) (234) (24) 48 1 2 1 5 2 1 5 1 4 6 4 1 (13)(24) (23) (12) 6 1 2 1 5 2 1 6 1 4 6 2 3 (13)(24) (14) (12) 6 End of execution.

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MOVES IN THE SQUARES GROUP WHICH FIX CORNERS. KEY: 1=L; 2=F; 3=R; 4=B; 5=U; 6=D (ALL 180 DEGREE TURNS). THE NUMBERS ON THE RIGHT GIVE THE PERMUTATION OF EACH SLICE. THE ORDER IN WHICH SLICES ARE CONSIDERED IS FB,UD,LR. FB SLICE: 1=LU, 2=LD, 3=RD, 4=RU. UD SLICE: 1=LB, 2=LF, 3=RF, 4=RB. LR SLICE: 1=FU, 2=FD, 3=BD, 4=BU. THE NUMBER ON EXTREME RIGHT IS THE QUANTITY OF SYMMETRICALLY EQUIVALENT POSITIONS PAGE 6: NO SLICE FIXED; TWO 4-CYCLES FB UD LR 1 2 1 2 1 2 1 2 4 1 5 6 (1324) (13)(24) (1423) 12 1 2 1 2 1 2 1 2 4 5 6 3 (1324) (14)(23) (1423) 24 1 2 1 2 1 3 4 5 2 3 4 5 (1234) (13)(24) (1324) 24 1 2 1 2 1 3 5 1 2 1 6 3 (1423) (14)(23) (1432) 24 1 2 1 2 1 3 5 1 2 5 1 6 (1423) (1324) (243) 48 1 2 1 2 1 3 5 1 2 6 3 5 (1423) (1423) (243) 48 1 2 1 2 1 3 5 1 5 1 6 4 (1423) (1432) (243) 96 1 2 1 2 1 3 5 1 6 3 5 4 (1423) (1234) (243) 96 1 2 1 2 1 3 5 2 5 1 2 6 (234) (1324) (1324) 96 1 2 1 2 1 3 5 2 6 3 4 5 (234) (1423) (1324) 96 1 2 1 2 1 5 2 1 4 5 4 3 (1342) (12)(34) (1423) 24 1 2 1 2 3 5 1 2 5 2 1 6 (13)(24) (1243) (1324) 6 1 2 1 2 3 5 1 2 6 4 3 5 (13)(24) (1342) (1324) 6 1 2 1 2 3 5 1 4 5 2 3 5 (14)(23) (1342) (1324) 12 1 2 1 2 3 5 1 4 6 4 1 6 (14)(23) (1243) (1324) 12 1 2 1 2 3 5 1 5 2 5 1 6 (1423) (124) (1324) 48 1 2 1 2 3 5 1 6 4 5 3 5 (1423) (132) (1324) 24 1 2 1 2 3 5 1 6 4 6 1 6 (1423) (143) (1324) 24 1 2 1 2 4 3 5 1 2 1 6 4 (1423) (12)(34) (1342) 24 1 2 1 2 4 3 5 1 5 2 1 6 (13)(24) (1243) (1342) 24 1 2 1 2 4 5 1 5 1 6 2 1 (13)(24) (1342) (1234) 24 1 2 1 2 5 1 5 2 3 5 4 1 (14)(23) (1432) (1234) 24 1 2 1 2 5 1 5 4 1 6 2 1 (13)(24) (1432) (1234) 12 1 2 1 2 5 1 5 6 2 3 5 1 (1423) (12)(34) (1234) 24 1 2 1 2 5 2 3 5 3 2 6 4 (1423) (1234) (14)(23) 24 1 2 1 3 5 1 5 1 2 3 4 5 (1234) (1342) (143) 96 1 2 1 3 5 1 5 1 6 3 4 1 (14)(23) (1342) (1432) 24 1 2 1 3 5 1 5 3 2 3 2 5 (1432) (1342) (124) 96 1 2 1 5 1 5 3 2 5 4 6 1 (1432) (124) (1234) 48 1 2 1 5 1 5 3 2 6 2 5 1 (1432) (143) (1234) 48 End of execution.

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MOVES IN THE SQUARES GROUP WHICH FIX CORNERS. KEY: 1=L; 2=F; 3=R; 4=B; 5=U; 6=D (ALL 180 DEGREE TURNS). THE NUMBERS ON THE RIGHT GIVE THE PERMUTATION OF EACH SLICE. THE ORDER IN WHICH SLICES ARE CONSIDERED IS FB,UD,LR. FB SLICE: 1=LU, 2=LD, 3=RD, 4=RU. UD SLICE: 1=LB, 2=LF, 3=RF, 4=RB. LR SLICE: 1=FU, 2=FD, 3=BD, 4=BU. THE NUMBER ON EXTREME RIGHT IS THE QUANTITY OF SYMMETRICALLY EQUIVALENT POSITIONS PAGE 7: NO SLICE FIXED; 2-CYCLE + 4-CYCLE. FB UD LR 1 2 1 2 1 2 1 3 5 2 4 5 (34) (13)(24) (1423) 24 1 2 1 2 1 2 1 5 2 4 5 3 (34) (14)(23) (1423) 24 1 2 1 2 1 3 2 5 2 1 2 6 (24) (13)(24) (1324) 24 1 2 1 2 1 3 5 1 2 3 5 1 (1423) (14)(23) (24) 24 1 2 1 2 1 3 5 1 3 5 1 4 (1423) (14)(23) (34) 24 1 2 1 2 1 5 1 2 5 4 1 6 (14)(23) (1342) (34) 24 1 2 1 2 1 5 1 3 2 5 4 6 (143) (1342) (34) 96 1 2 1 2 1 5 1 3 2 6 2 5 (143) (1243) (34) 96 1 2 1 2 1 5 1 4 5 4 3 5 (13)(24) (1243) (34) 24 1 2 1 2 1 5 1 5 4 5 3 5 (1423) (124) (34) 48 1 2 1 2 1 5 1 5 4 6 1 6 (1423) (234) (34) 48 1 2 1 2 1 5 1 6 2 5 1 6 (1423) (132) (34) 48 1 2 1 2 1 5 1 6 2 6 3 5 (1423) (143) (34) 48 1 2 1 2 1 5 2 1 2 6 2 3 (23) (12)(34) (1423) 24 1 2 1 2 4 1 5 1 2 1 6 2 (1324) (12)(34) (14) 24 1 2 1 2 4 1 5 1 5 4 3 5 (13)(24) (1243) (14) 24 1 2 1 2 4 1 5 2 3 5 2 5 (1234) (23) (12)(34) 24 1 2 1 2 4 1 5 2 5 2 5 3 (1234) (24) (12)(34) 24 1 2 1 2 4 3 5 2 5 2 6 3 (24) (1234) (13)(24) 24 1 2 1 2 4 5 1 5 1 5 2 3 (12)(34) (14) (1432) 24 1 2 1 2 4 5 1 5 3 5 4 3 (13)(24) (1342) (13) 24 1 2 1 2 4 5 2 5 1 4 5 1 (24) (1324) (14)(23) 24 1 2 1 2 4 5 2 5 6 1 2 5 (1234) (12)(34) (12) 24 1 2 1 2 4 5 2 5 6 3 2 6 (24) (14)(23) (1324) 24 1 2 1 2 5 1 3 2 5 1 4 5 (132) (12) (1423) 96 1 2 1 2 5 1 3 2 5 3 4 6 (143) (1423) (34) 96 1 2 1 2 5 1 3 2 6 1 2 5 (143) (1324) (34) 96 1 2 1 2 5 1 3 2 6 3 2 6 (132) (34) (1423) 96 1 2 1 2 5 1 5 1 2 4 6 4 (1324) (13) (142) 96 1 2 1 2 5 1 5 1 6 4 5 6 (12) (1234) (142) 96 1 2 1 2 5 1 5 3 2 4 6 2 (34) (1432) (134) 96 1 2 1 2 5 1 5 3 6 2 5 6 (1423) (24) (134) 96 1 2 1 2 5 1 5 6 4 1 5 3 (12) (13)(24) (1432) 24 1 2 1 2 5 2 1 5 3 2 6 2 (34) (1432) (13)(24) 24 1 2 1 2 5 2 5 2 6 1 4 1 (23) (1423) (14)(23) 24 1 2 1 3 5 1 2 1 6 2 5 6 (34) (14)(23) (1342) 24 1 2 1 3 5 1 2 3 6 4 5 6 (1324) (13)(24) (14) 24 1 2 1 3 5 2 5 2 5 3 5 6 (1234) (13) (14)(23) 24 1 2 1 5 1 5 1 2 4 5 4 3 (12)(34) (1243) (13) 24 1 2 1 5 1 5 1 2 5 4 5 3 (1234) (124) (13) 96 1 2 1 5 1 5 1 2 6 2 6 3 (1234) (143) (13) 96 1 2 1 5 1 5 2 5 2 3 6 3 (12) (132) (1432) 96 1 2 1 5 1 5 2 5 4 3 5 3 (1423) (124) (13) 96 1 2 1 5 1 5 2 6 2 1 6 3 (1423) (143) (13) 96 1 2 1 5 1 5 2 6 4 1 5 3 (12) (234) (1432) 96 1 2 1 5 1 5 3 6 2 5 6 1 (14)(23) (14) (1234) 24

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A DETAILED EXAMPLE OF THE 52-MOVE STRATEGY (UFL)+ (URF)- (UBR)+ (ULB)- (LF)* (FR)* (RB)* (BL)* First decide on a coordinate system for the Cube (i.e. decide which colours are L,R,F,B,U,D); then get it into the above position. Stage 1. There are 4 bad edge-pieces, namely in positions LF,FR,RB,BL. Manoeuvre these to the U-face by FLR'D2B2. Then the move U corrects them. Summary of Stage 1:- FLR'D2B2U (6 moves). Stage 2. The LR-slice edge pieces are now in positions LD,FD,RD,BD. Manoeuvre these to the UD-slice by F2D2LR'F. Now taking the corner positions in order (as in the diagram in Stage 3 instructions), the respective twists of the pieces in these positions are 0,2,0,2,0,0,1,1. This combination of twists is not given in Stage 2 tables, but a 180° rotation about the LR-axis followed by reflection in the LR-slice transforms this to 2 2 0 0 0 1 0 1, which is in the tables. The move given is F LFL2F'LF2BL2, and transforming this by the above (involutory) symmetry gives B'R'B'R2BR'B2F'R2. Therefore we perform the inverse of this move, after which L and R faces have L and R colours on them only. Summary of Stage 2:- F2D2LR'FR2FB2RB'R2BRB (14 moves) Stage 3. The positions where corners are out of orbit are numbers 1,2,5,8. The preliminary instructions for this stage instruct us to perform L'U2. For the remainder of this stage alter the coordinate system so that the original D-face faces you and the original F-face faces upwards. In this new coordinate system the positions where corners are out of orbit are 1,5. The permutation of corners is (1357)(24). Multiplying this on the right by (15)(24) gives (13)(57) which is a permutation of corners in G3. Therefore we must refer to page 7 of the Stage 3 tables. The edge- pieces of the FB-slice are in positions 3,4,5,8. The tables give us LF2L'U2LF2R2F2B2R, or LD2L'F2LD2R2D2U2R in the original coordinate system. Perform the inverse of this move. Summary of Stage 3:- L'U2R'U2D2R2D2L'F2LD2L' (12 moves). Stage 4. The corners are restored by L2R2(original coordinates). Looking at edge-pieces, as no slice is fixed and there is a 2-cycle and a 4-cycle, we refer to page 7 of the Stage 4 tables. The only entries where the correct arrangement of pieces is permuted and the 4-cycle is of the correct type are (143) (1342) (34) and (143) (1243) (34). If we hold the Cube with the original R-face facing the operator, and the original B-face uppermost, we find we have the second of these permutations. Therefore perform the inverse of the move given, i.e. perform B2R2F2R2D2U2B2U2R2U2R2U2in original coords. Summary of Stage 4:- L2R2B2R2F2R2D2U2B2U2R2U2R2U2(14 moves). Total number of moves required: 6+14+12+14 = 46. (45 with cancellation) M.B.T.