[Math] Rubik’s cube function

functionsrubiks-cube

I'm thinking in a function and if it's possible to solve that.

I have been playing with the cube using the following move: $R U L' U'.$

I notice that the cube solves itself with a certain number of moves: 28 moves to $2\times2\times2$ and 112 to $3\times3\times3.$ (If the cube already is solved).

Then I'm trying to create a formula to calculate the number of moves for the another cubes like $4\times4\times4, 5\times5\times5, 6\times6\times6\dots$

Since:

$x \rightarrow y$

$1 \rightarrow 0$

$2 \rightarrow 28$

$3 \rightarrow 112$

$4 \rightarrow z$

Where x is the number of the cube $(2 = 2\times2\times2, 3 = 3\times3\times3 \dots)$ and y is the amount of moves, I came up with two formulas: 28*$((x-1)^{(x-1)})$ and 28*$(x-1)^2$. Thus, the value for $z$ could be 252 or 756,

My questions are:

  1. Are any of these formulas correct? If so, which one?
  2. Can be my reasoning corret about the formulas?
  3. If I'm wrong, answer me why!

Best Answer

The answer for a $4 \times 4 \times 4$, $5\times 5\times 5$, and so on cube will continue to be $112$.

Here's why. Imagine that you take an $n \times n \times n$ cube and, on each face, glue the pieces in the middle $(n-2) \times (n-2)$ square together, so they cannot be separated. Similarly, glue the middle $n-2$ pieces along each edge together. What you have is no longer an $n \times n \times n$ cube, but a $3 \times 3 \times 3$ cube in which some of the pieces are much larger in size. But it still operates exactly like a $3 \times 3 \times 3$ cube and affords exactly the same twists.

Your repeated RULU' move is acting on the $n \times n \times n$ cube in a way that gluing the pieces as above doesn't forbid. So whatever happens on the $n \times n \times n$ cube for $n>3$ is exactly the same as what happens for the $3 \times 3 \times 3$ cube.

The only reason that the $2 \times 2 \times 2$ cube is an exception is that here, there are no "edge" pieces at all. So if a sequence of moves in the $3 \times 3 \times 3$ cube scrambles (or flips) the edges but leaves the corners fixed, then on the $2 \times 2 \times 2$ cube, it does nothing, and this exactly describes what $28$ iterations of RULU' do.

Related Question