A conventional Rubik's Cube with six coloured faces has forty-three quintillion permutations, but it does have a single solution.

Ever the puzzler and mind confuser, the venerable De Selby (of Third Policeman fame) was not satisfied with this number, large as it was, and set himself thinking about
creating a new, more devious cube. In fact it would be one without any satisfactory conclusion, regardless of the amount of twiddling and fiddling and algorithms.

This is because each of De Selby's Cubes has no solution. The reason for this lies in the construction and rationale of the cubes. Every single one of the 54 facets on all six of their sides has a different colour randomly selected from the within the pantone colour range.

This means that every complete cube is totally unique and no cube ever has a "solution".

"Every single one of the 54 facets on all six of their sides has a different colour randomly selected from the within the pantone colour range."
It seems likely that, if you made enough of them and they're truely random, at least some would be solvable.

//It seems likely that, if you made enough of them and they're truely random, at least some would be solvable.// The point is that there is no natural "rest" position, therefore there is no "solution".

For the record, I have a 2x2x2 Rubik's Cube, the regular 3x3x3 version, the larger 4x4x4 version known as "Rubik's Revenge", and the relatively new 5x5x5 version (just got it as a present).

It might be relevant to this Idea that back in the early 1980s when it was just a hobby/learning thing, I wrote a computer program to simulate a 4x4x4 cube (I didn't know if it was possible to be built; the real cube hit the market about 6 months after I started; and I didn't finish for another year or so). When I got done, if you had a real Rubik's Revenge scrambled up in your hand, you could feed the pattern of colored spots into the computer, and it could solve the puzzle (displaying about one twist per second). The sequence of moves could be reviewed one-at-a-time, allowing you to apply the solution to the physical cube.