From: Chas F Brown <cbrown@cbrownsystems.com>
Subject: Re: 3d tic-tac-toe
Date: Tue, 09 Jan 2001 16:02:08 -0800
Newsgroups: sci.math
Summary: Magic Cubes

Bill Taylor wrote:
> 
> Lewis Mammel <lhmammel@ameritech.net> writes:
> |>
> |> Here's something that just occurred to me. I suppose it's commonly
> |> noted that the 8 central cells are strategically equivalent to
> |> the 8 corners in 4x4x4 tic-tac-toe.
> 
> Yes, it is very commonly known and in various books, I guess most 4TTT
> folk discover it; I know I did as a child, (Hello Pertti!), as did many
> others I'm sure.
> 
> It is remarkably similar to the reversability between corners and centres
> in 4x4 magic squares!  No doubt if there are 4x4x4 magic cubes it will be
> an even closer analogy.
> 
> -------------
> 
> Which makes me think about magic cubes.   Do these exist?  It seems an
> obvious generalization that must've occurred to thousands of other folks,
> but I don't recall ever seeing it in print.
> 
> Anyone know?
> 

Typing "magic cube" into google.com gives a number of sites with magic
cubes, hyper cubes, etc. 
An overview of their construction (along with refernce to some books and
articles) is given at:

http://www.inetworld.net/~houlton/cube.html

In particular, this site:

http://makoto.mattolab.kanazawa-it.ac.jp/~poyo/magic/cube/

has examples of magic cubes up to 25x25x25, and at

http://makoto.mattolab.kanazawa-it.ac.jp/~poyo/magic/hypermagiccube/3x3x3x3x3x3.html

he has a six-dimensional magic hypercube of order 3, with all sums comin
to 1095 (the hardest part is visualizing it!).

Cheers - Chas

> In the case of 3x3x3 it seems there should be "less chance" of one existing
> than for the 3x3 square.  The 3x3 square has to get 8 totals equalizing,
> and has 9 numbers to choose, giving one d.f. to play with.
> 
> But the 3x3x3 cube has far more lines to equitotalize, than the only 27
> numbers to do it with.  Even if we drop the diagonals, and just require
> orthogonal totals to be equal, there are still 27, so this could maybe
> JUST be possible.
> 
> Orthogonal restriction would help out a lot with larger cubes - there are
> only 48 ortholines in a 4x4x4 and a huge 64 numbers to choose. So there
> should be TONS of these.  But with diagonals there are 76 lines - looks bad.
> 
> But for 5x5x5 there are 109 lines including diags and 125 numbers to choose.
> This should proviude a rich harvest again!
> 
> Is anything known about all these?
> 
> -----------------------------------------------------------------------------
>    Bill Taylor                             W.Taylor@math.canterbury.ac.nz
> -----------------------------------------------------------------------------
>         I don't live on the edge, but sometimes I go there to visit.
> -----------------------------------------------------------------------------

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