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Here is some more Cube Theory:

On the standard Rubik's Cube

Using < R, L, F, B, U > to generate D1 Let A = R1 L3 F2 B2 R1 L3, then A U1 A = D1 A U1 A = R1 L3 F2 B2 R1 L3 U1 R1 L3 F2 B2 R1 L3 (17 q, 13 q+h) Using < R2, L2, F2, B2, U2 > to generate D2 Let A = R2 F2 B2 L2, then A U2 A = D2 A U2 A = R2 F2 B2 L2 U2 R2 F2 B2 L2 (18 q, 9 q+h) Slice group pattern, 6 spot, 4 slice moves p1 = (F1 B3) (L1 R3) (U1 D3) (F1 B3) (8 q) Slice group antipode, 6 spot + pons asinorum, 6 slice moves p2 = (F2 B2) (T1 D3) (F1 B3) (L3 R1) (T1 D3) (12 q) p2^6 = I p7a Cube in a cube U2 F2 R2 U3 L2 D1 (B1 R3) ^3 + D3 L2 U1 (15 q+h, 20 q)

if A = U3 L2 D1, then let A' = inverse of A

p7a = U2 F2 R2 + A + (B1 R3) ^3 + A'

Or if we want something more symmetric, there is Mike Reid's...

p7b Symmetric Maneuver (R3 U1 F2 U3 F3 L1 F2 L3 F1 R1 C_X ) ^ 2 (20 q+h , 24q)

One might even call this maneuver to be "cyclic decomposable".

Even the first half of this sequence generates an interesting pattern.

It would appear that using symmetric maneuvers does not ensure minimal

q or q+h turns.

Perhaps p7a is simpler in terms of notational expression. p7a is

how I actually do "Cube in a cube" in real cubing. Note also that

p7a uses all 6 generators and p7b uses <U, L, R, F> and there may

be a tighter symmetric "Cube in a cube".

Mike, have you tried using the pattern generated by the first half under your Kociemba algorithm for q turns?? ----------------------------------------------------------------------

Megaminx (platonic dodecahedron) 12 faces, 20 corners, 30 edges

tetrahedron = 4 axis cube = 3 axis dodecahedron = 6 axis

In constructing a dodecahedron, build a bottom, place the 5 adjacent

faces to form a bowl. The top edges now form a skew decagon. Build

another bowl and connect the two bowls together to form a dodecahedron.

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Recall that the Halpern-Meier Tetrahedron has 3,732,480 states.

In this count we consider the 4 centre pieces immobile.

The picture H-M tetrahedron has 3,732,480 * 3^3 = 100,776,960 states. In essence, we can rotate any 3 centres at will, the 4th is forced. That number may seem familar to some of the more fanatical cubists, has the picture Skewb has 3,149,280 * 2^5 = 100,776,960 states! Additionally the possible rotations of the centres of the H-M tetrahedron are (), (+ -), (+++) , (---), (-++-), (+--+), (+++-), (--+-) # Order (tetra, r * d) = 45 OR (r+ d+)^45 = I Note that (r+ d+)^45 is still the identity, even on the picture H-M tetrahedron, as 45 is divisible by 3. ------------------------------------------------------------- -> Mark <-