Message #562

From: David Smith <>
Subject: 4x4x4x4 Formula found
Date: Sun, 14 Sep 2008 17:37:34 -0700

Hello everyone,
I just wanted to let everyone know that I have finished counting an upper bound for the
number of permutations of an nxnxnxn Rubik’s Cube, the result is here.  Despite the
appearance of the formula, it was not very hard to discover, thanks to Eric Balandraud and
his page on the MagicCube4D website.  Without that page, the result would have been
much harder to deduce.  Unfortunately, the 5x5x5x5 formula listed on that page is
incorrect, but I believe it is just a typo, as the two errors are obvious and are in
conflict with the same principles described earlier on the page.
Now I will try to find the same formulae for the n^4 supercube (a Rubik’s Cube in which
each cubie is uniquely identifiable in any position or orientation), super-supercube (a
supercube in which there is a series of supercubes inside of it - (n-2)x(n-2), (n-4)x(n-4),
etc.), the n^5 cubes, and eventually the n^k cubes.  Given the nature of the n^4 normal
cube, I imagine that these formulae will be extraordinarily complex.  Figuring these out
will be much more difficult because I will have to determine the laws of the cubes myself,
rather then having them available via Eric’s page.  His page not only was very helpful
for the n^4 cube, but by verifying the cube’s properties, I have a better foundation for
discovering the properties of higher-dimensional cubes and the "super" variants.
Thank you, Eric!
I hope everyone is doing well.  Thanks for the book recommendations, Melinda!  I agree
with you completely on the matters regarding higher-dimensional eyes and legs.
Many thanks to Jenelle and Guy for raising these intriguing questions.  By the way, I may
also try to determine an upper bound for the number of moves required to solve
MagicCube4D from any position.  Although I think exploring this question would have a
more direct impact on and be more of an interest to the group, I am less confident I can
provide any answers here.
Once again, I would like to thank Roice for his kindness in hosting my papers and
results on his website.  I look forward to sharing my future results with you.
All the Best,