From: Chas F Brown 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 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. > ----------------------------------------------------------------------------- --------------------------------------------------- C Brown Systems Designs Multimedia Environments for Museums and Theme Parks ---------------------------------------------------