The Gear Octahedron puzzle actually has $1,327,104$ states. On my website I have a page about the Gear Mastermorphix, which is an equivalent puzzle (same mechanism, but with a tetrahedral outer shape). On that page there is an explanation as to how that number comes about:
$4!$ permutations of the triangular face centres (one of them is held steady and ...
I found a super-scramble that has the folowing properties:
No squares of the same color touch orthogonally or diagonally
Each color is present on all sides 1 or 2 times
All combinations of 4 colors are present at an intersection of 4 squares. Possibly over the edge.
I am still missing
No squares of the same color touch diagonally over an edge.
Here it ...
You should start with a list of the small cubies (red-blue-white corner, green-yellow side, orange center, etc), and make a random configuration with the one required side in place.
Then, you must guard against the three possible parity errors that can cause an invalid scramble:
The total amount of twisting that the corners need must be sum up to ...
This is in principle exactly the same as just solving the cube, but in practice it's much harder on the brain. Here's a way to do it that's kinda cheaty but demonstrates the equivalence.
You're starting with a solved cube. For each facelet of the cube, figure out where it needs to go. Look at what colour is in that place right now, on the solved cube. ...
Set up the pattern you want. Solve it, writing down the moves you used.
You now have a solved cube. If you undo the move sequence you wrote down, you will get the original pattern. In other words, if you invert the written move sequence (reverse the order of the sequence, and the turning direction of each move) you get a move sequence that generates the ...