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Additional Notes on Squares Group Patterns ------------------------------------------ Note that p80a, p99a and p108a are 2 DOT patterns, all of the form U1 (swap edges & corners in U and D faces) D3 or U1 (swap edges & corners in U and D faces) D1 or D1 (swap edges within U and edges within D) D3 P66a alternate method F2 R2 U2 F2 R2 U3 D3 B2 L2 F2 B2 U1 D1 (13) p67a alternate method F2 R2 F2 U3 D3 L2 B2 D2 L2 B2 U1 D1 B2 (13) p80a alternate method U1 F2 R2 L2 U2 D2 F2 U2 D3 (9) p99a alternate method U1 R2 F2 B2 U2 D2 R2 D1 (8) P100a alternate method F2 U2 D2 F2 R2 L2 D1 F2 R2 L2 B2 U1 (12) p108a alternate method R2 F2 B2 L2 D1 R2 U2 R2 L2 U2 R2 D1 (12) p130a alternate method F2 R2 F2 B2 U1 D1 F2 R2 D2 F2 L2 U3 D3 (13) p133a alternate method R2 U1 F2 R2 L2 U2 D2 F2 U2 D3 R2 (11)

A) In general, any sequence of half turns which swaps edges and

corners in the U and D faces can be sanwiched between a single quarter

turn of U and a single quarter turn of D. Such a process would lead to

a square's group position.

B) Furthermore, any sequence of half turns which swaps edges within U

and

edges within D can be sanwiched between a single quarter turn of U or D

and a single quarter turn of U or D. Once again, such a process would

lead to a square's group position.

Here is an example of a position which takes over twice as many half

turns as full group moves:

L2 T2 L2 T2 L2 T2 F2 L2 T2 F2 T2 R2 B2 (13) U1 F2 R2 L2 B2 D1 (6)

As discussed in point A above, sequences which move all elements of

the U face to the D face and also move all elements of the D face

to the U face (excepting the centres naturally) appear as 2 DOT

patterns on the cube. This makes sense, as the initial quarter

turn in process p80a must be balanced by another quarter turn.

Since all of the elements subjected to the quarter turn are now

in the D face, we must turn that face a quarter turn to remain

in the squares group.