Every now and then, scrambling my Rubik's cube I find it in a state where every side has at least one tile in each of the six colors. For example here's a scramble I just did (obviously put through a solver after-the-fact to get an optimized algorithm):
F2 U' L2 R2 B2 U R2 D U2 B2 U R2 F' R' B2 D' B' L' D2 B' F R2
This results in the following configuration (where U is white and F is green):
There are absolutely no other special properties about these scrambles that I can think of, other than the fact that every color appears at least once, but never more than three times, on each side. In particular, any number of tiles of the same color may be adjacent to one another, and it doesn't matter to me "how" randomized it is.
These cases don't seem particularly uncommon, as I stumble upon them once every few dozen solves. How many such permutations that are solvable exist?
Also, are there any legal permutations where the same color appears four times on the same side (leaving one for each of the remaining five colors), and why/why not? I'm surprised I haven't come across any yet — I figured I would have come across a few of them if they did in fact exist.