Two questions on the 2x2x2 Rubik's cube

Factual Questions
1

  1. What are the odds that, after a scramble, no two colors are paired up next to one another?

  2. What is the most turns (double twists = one turn) ever required to get an entire side one color? (any color, rest of cube doesn’t matter)

2

IANA mathematician, but I think think the maximum number of moves required for the 2x2 cube would be very small. For the original 3x3 cube, there were two or three solution booklets published which demonstrated that twelve or so moves would always be sufficient to solve the puzzle. (Or maybe it was sixteen, IIRC with each move being a 90-degree twist.)

3

Define your first question. My assumption is that you mean no two of the same color are next to each other.

If that’s the case, then it’s definitely possible. My finite is rusty but… it should be like,

(8! x 3^7) / 24, which is like… 3674160

Some of those will have no two touching of the same color.

As for your other question. No more than like 5 or 6 if you’re not caring how their secondary sides line up to keep working the cube properly.

4

It’s 20 (where a move can be 90 or 180 degrees).