# Diversified Rubik's Cube [duplicate]

I have, cleverly I hope, shuffled my Rubik's Cube for a while. I could not find a position where, for each faces, all colors appear 3 times or less. I always had a or several faces with at least 4 of the same colors.

This puzzle is to determine what is the minimum number of colors $$X$$ such that there exists a Rubik's Cube position which for all faces, all colors appear less than $$X$$ times.

Each color appears $$9$$ times total, so an average of $$9/6$$ times per face, which implies a lower bound

$$X \ge \lceil 9/6 \rceil = 2.$$

Without loss of generality, color $$c$$ appears at the center of face $$c$$. Here is such an assignment of colors to faces, where each row or column sums to $$9$$ and the diagonal is positive:

$$\begin{matrix} \text{color}\backslash\text{face} &1 &2 &3 &4 &5 &6 \\ \hline 1 &1 &2 &2 &2 &0 &2 \\ 2 &2 &2 &0 &2 &2 &1 \\ 3 &2 &0 &2 &1 &2 &2 \\ 4 &1 &1 &1 &2 &2 &2 \\ 5 &1 &2 &2 &2 &2 &0 \\ 6 &2 &2 &2 &0 &1 &2 \\ \end{matrix}$$

Not sure whether these counts are actually physically realizable on the cube.

I played with an actual cube and got

$$X \le 3$$:

(The top right corner is orange.)

• Do you choose to upvote only the best of the best questions?
– JLee
Jul 30, 2022 at 1:29
• @JLee I haven't kept track, but I probably upvote answers more often than questions. Why do you ask? Jul 30, 2022 at 1:36
• Just something I noticed. You provide very high-quality answers. I always enjoy reading them & learning. So thanks for that. Often I see that there are zero upvotes on a question/puzzle that you answer, & I assume, "He thought this puzzle/question was worth answering (many times in great detail with lots of time spent) but not worth an upvote?" To me, the vast majority of the time, if I am going to spend more than a tiny bit of time on something, my upvote of the puzzle is a "thank you" for providing me with something interesting to think about. But that's just me, and everyone's different.
– JLee
Jul 30, 2022 at 11:07
• I see. I took some time just now to upvote many questions I have answered, including at least one of yours. 😊 I hit my vote limit for the day but will keep this in mind. Jul 30, 2022 at 14:38