How Many Permutations Can a 1000x1000x1000 and 5-Dimensional Rubik's Cube Have?

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Discussion Overview

The discussion centers on the number of permutations for a 1000x1000x1000 Rubik's cube and a 5-dimensional Rubik's cube. Participants explore mathematical formulations and concepts related to permutations in higher-dimensional cubes, including references to existing literature and personal experiences with mathematical tools.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant expresses difficulty with large numbers and seeks the number of permutations for both a 1000x1000x1000 cube and a 5-dimensional cube, referencing existing formulas and resources.
  • Another participant provides a link to a sequence that may assist in understanding the permutations.
  • A different participant suggests that the permutations form a finite index subgroup of a direct product of wreath products, similar to the 3x3x3 cube, and encourages others to work through the problem independently.
  • One participant shares their struggle with the complexity of the mathematics involved and expresses a desire for straightforward formulas for both NxNxN and NxNxNxNxN cubes.
  • Another participant mentions the importance of understanding wreath products and permutation groups, recommending a book that covers the computation of permutations for the 3x3x3 cube and hints at the complexity of extending this to higher dimensions.
  • A participant describes their experience converting a Maple script to Java for calculating permutations but admits to confusion over variable definitions in the context of the formula for a 3-D Rubik's cube.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the exact formulas for the permutations of the specified cubes. Multiple viewpoints and approaches are presented, indicating ongoing uncertainty and exploration of the topic.

Contextual Notes

Participants express varying levels of mathematical understanding and familiarity with advanced concepts such as wreath products and permutation groups, which may affect their ability to engage with the discussion. There are references to specific mathematical tools and programming languages that may not be universally accessible.

MALON
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Rubik's cube permutations

I suck with really big numbers, so that's where you guys come in :)

I basically want to know the number of permutations a 1000x1000x1000 Rubik's cube has, as well as a 5x5x5x5x5 Rubik's cube. Yes, a 5-dimensional one.

I've been reading these a formula's, and they aren't that comlicated, they just involve huge numbers, and that's where my brain shuts down.

You can read about 3x3x3, 4x4x4, and 5x5x5 permutations on Wikipedia. I will provide links for all. I will also provide a link about the anatomy of an n-dimensional Rubik's cube as well as the permutations for a 3x3x3x3, 4x4x4x4, and 5x5x5x5 Rubik's cube, similar in the same manner Wikipedia does.

3x3x3: http://en.wikipedia.org/wiki/Rubik's_Cube[/PLAIN]

4x4x4: http://en.wikipedia.org/wiki/Rubik's_Revenge[/PLAIN]

5x5x5: http://en.wikipedia.org/wiki/Professor's_Cube[/PLAIN]

n-dimensional Rubik's anatomy: http://www.gravitation3d.com/magiccube5d/anatomy.html

3x3x3x3, 4x4x4x4, 5x5x5x5: http://www.superliminal.com/cube/permutations.html



Thank you guys for reading! Hope you enjoy computing :)
 
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Here's a place to start:
http://www.research.att.com/~njas/sequences/A075152
 
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It should be a finite index subgroup of a direct product (running over the orbits of the "cubies") of wreath products, as for the 3x3x3 cube. Before looking up the answer, I suggest you try working it out yourself following the excellent explanation of the 3x3x3 case at http://unapologetic.wordpress.com/category/rubiks-cube/
 
Sorry Chris, the math there is beyond me. I've tried figuring this out many times before. I enjoy CRGreathouse's post because that gave me a formula in which to figure out an NxNxN cube, although I'm not sure how to run the script in Maple. I used Maple and converted the script to Java which is far easier for me to read and it compiles after a tweaking session, but I don't know what variables do what :\

At least I'm a bit farther. I was just hoping someone could say "here's a formula for an NxNxN and one for NxNxNxNxN" or a formula for calculating the exact figures (1000x1000x1000 and 5x5x5x5x5). Maybe even a formula for N-sided I-dimensional. Apparently it's not that easy.

I always make the assumption that because I suck at math, everyone else is amazing at it and this is child's play to them. Figuring out these formula's is like me trying to comprehend Graham's number.

Thanks for your effort so far though!
 
C. R. Greathouse?

This isn't hard if you know what a wreath product is. That isn't hard if you know what a permutation group is. If you're curious, try Neumann, Stoy, and Thompson, Groups and Geometry, Oxford University Press, 1994 which is a really readable and wonderful book with very few prerequisites. The 19th and last chapter computes the number of elements in the group of the 3x3x3 Rubik's cube; a similar computation is given by John Armstrong in the webpage I cited. Once you understand this, you can make a start on the 4x4x4x4 cube and so on.

(Well, it might help to know something about SO(4) and so on in order to make sure you have the right generators of the permutation group whose size we are trying to compute--- you didn't say but I assume you are trying to compute the nxn..xn analog of the group of permutations of the set of "facelets" in the 3x3x3 cube obtained by iterating quarter turns of the six faces, so six generators.)
 
MALON said:
I used Maple and converted the script to Java which is far easier for me to read and it compiles after a tweaking session, but I don't know what variables do what :\

The formula is for the number of permutations in a 3-D Rubik's cube. n is the number of sides to the Rubik's cube, and the large expression between fi and end is the number of permutations in total. A through G are defined in the program.