12

In this MO question, user Martin Brandenburg asks about God's number for $n \times n \times n$-cubes for $n>3$. Here, God's number $g(n)$ was defined as the smallest number $m$ such that every realizable arrangement of the $n \times n \times n$-cube can be solved within $m$ moves.

We could generalize the notion of God's number to $n^{k}$-cubes, where $k >3$ is the number of dimensions. Define $g_{k}(n)$ as the smallest number $l$ such that every realizable arrangement of the $n^{k}$-cube can be solved within $l$ moves.

Whereas Mr. Brandenburg's question pertains to three dimensional Rubik's cubes ($k=3$), I wonder what is known about higher God's number for higher dimensional sequential move puzzles.

Questions:

  1. Is $g_{4}(2)$ known? And what about $g_{4}(3)$?
  2. What is known about the asymptotic value of $g_{k}(n)$?
Max Muller
  • 4,505
  • 1
    A relevant Reddit thread: https://www.reddit.com/r/Cubers/comments/arxuuh/4d_cube_gods_number/ – Max Muller May 08 '22 at 17:08
  • 5
    Could the downvoter please explain his/her motives? – Max Muller May 08 '22 at 22:21
  • Also: whence the vote to close? – Max Muller May 09 '22 at 20:51
  • 14
    I don't know about the closing vote, but certainly "God's number" is an irritating terminology (I'm aware you're not the first to use it). – YCor May 16 '22 at 09:32
  • If $k=3$, then an $n_k$-dimensional cube is $n^3$-dimensional, not three dimensional. The question confounds the dimension of the cube with the number of cubelets. Please decide on a consistent system of nomenclature, and edit the question accordingly. – Gerry Myerson Jun 25 '22 at 06:49
  • 2
    One might also ask first, what exactly is, say, a four-dimensional Rubik cube? Are we to fix it by "rotating" its three-dimensional "faces"? What sorts of rotation are allowed? – Gerry Myerson Jun 25 '22 at 06:54
  • 2
    @GerryMyerson Re your first comment: you're right - just now I edited the question to make the nomenclature consistent. As for your second question/comment: did you take a look at the following wiki article and references therein already? https://en.wikipedia.org/wiki/N-dimensional_sequential_move_puzzle – Max Muller Jul 01 '22 at 16:24

0 Answers0