A couple of months ago I was forced to remove my 3x3 cube simulation, The Cube, from the Apple App Store by Rubik Brands who complained that the app violated their Trademark.
Rubik Brands trademark was overturned in the EU. Rubik Brands asserts a 3D trademark on the arrangement of 27 cubes in...

Yes, when you are solving in stages you're dealing in cosets. One needs to take an arbitrary element of coset A to any member of coset B, say. One needs an optimal solver set up for the particular goal coset. Kociemba's Cube Explorer may be set up for problems such as that. I don't know...

Doing the math, if you have a cube state A which you want to take to B:
S * A = B
S * A * A'= B * A'
S = B * A'
So you need to solve both the starting state and the goal state. You then apply the generator for the goal state to the inverse of the starting state. An optimal solution of...

Here are three methods for manipulating the Up layer using conjugation. They're not optimal but they're easy to remember.
Three corner cycle
F D' F'
U
F D F'
U
F D' F'
U2
F D F'
Three corner cycle with corner twist
F D' F'
U
R' D' R
U
R' D R
U2
F D F'
Edge-corner three cycle
L' F L
U
L'...

I extended the above analysis to the Quarter Turn Metric and the Slice Turn Metric.
(I'm done trying to use the table editor here. You'll have to copy and paste into a spreadsheet if you're interested. The columns are the same as above.)
QTM
6 192 8 8
8 1,536 64 64
10...

I knocked together some code today and enumerated the set of cube states with five unsolved edge cubies. As it turns out the above calculation is not valid. There are 462,528 cube states with 5 unsolved edges not 712,800. The latter number includes a lot of duplicates of states with less than...

You are correct. My calculation includes cases which are not 5 cycles. There are three cycles + two cubies flipped in place, etc. The requirement to preserve edge orientation is ambiguous since that depends on how one defines edge orientation.

I did some back of the envelope calculations:
The number of sets of 5 edge cubies:
12! / 7! = 95,040
Each of these edge sets may be permuted:
5! / 2 x 2^4 = 60 x 16 = 960
giving:
95040 x 960 = 91,238,400 edge permutations of five or fewer cubies.
similarly
12! / 8! x 4!/2 x 2^3 =...

Another definition of edge flip counts the number of quarter turns required to return the edge piece to the solved position and orientation. Even flip requires an even number of q-turns. Odd flip requires and odd number of q-turns.

I chose the above macros in the context of what is the minimum a person needs to know in order to solve the cube. You can tell a novice that all he has to learn is these five macros and he can solve the cube simply by repeatedly applying them.

I have considered the problem of solving the cube using a small set of primitive macros such as:
A tri-corner swap
F' R L F' R F L' F' R' F R' F
A tri-edge swap
F' U F' U' F' U' F' U F U F2
A tandem edge+corner swap
R U2 F' R' F U' F' R F U' R' U'
A corner twist
F R U F U' B U F' U' R'...

The positions a 3x3x3 may be placed in by rotating the faces may be represented as a mathematical group. The size of this group is 43,252,003,274,489,856,000 and may be calculated in the manner shown in the video. This is the "real" size of the group.
When one is performing "god's algorithm"...

If by Picture cubes you mean 3x3x3 cubes where the orientation of the center cubies matter, then the diameter is not known but it is at least 24f. Superflip requires 24 face turns: R L' U R U F R' L2 U F D' F2 R L' B2 D B' U' L R2 B' U' L' U'