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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 $g_{k}(n)$?
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    $\begingroup$ A relevant Reddit thread: reddit.com/r/Cubers/comments/arxuuh/4d_cube_gods_number $\endgroup$
    – Max Lonysa Muller
    Commented May 8, 2022 at 17:08
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    $\begingroup$ Could the downvoter please explain his/her motives? $\endgroup$
    – Max Lonysa Muller
    Commented May 8, 2022 at 22:21
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    $\begingroup$ 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). $\endgroup$
    – YCor
    Commented May 16, 2022 at 9:32
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    $\begingroup$ 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? $\endgroup$
    – Gerry Myerson
    Commented Jun 25, 2022 at 6:54
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    $\begingroup$ @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? en.wikipedia.org/wiki/N-dimensional_sequential_move_puzzle $\endgroup$
    – Max Lonysa Muller
    Commented Jul 1, 2022 at 16:24

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