Newsgroups:     rec.puzzles,sci.math
From:           umatf071@unibi.hrz.uni-bielefeld.de (sio)
Subject:        Re: Tiling problem
Date:           Tue, 9 Feb 93 21:28:17 GMT
Organization:   Universitaet Bielefeld

The last tetracube tiling problem: (Update of a previous posting)

 ______
|\     \               Is it possible to tile a 3*2n*2m box only
| \_____\              with tetracubes shown left?    
| |     |____
|\|     |    \         All other (hyper-) box tiling problems with
| *_____|_____\        only alike tetracubes are solved. See:
| |\     \    |         A. L. Clarke, Packing Boxes with Congruent Polycubes,
 \| \_____\   |         J. of Recreational Mathematics 10 (1977/78) 177-182
  * |     |___|
   \|     |
    *_____|

                 You can show 2 | nm.
                 According to my computations n and m had to be greater then 8.

The 3*4*Z is tileble:
    build two times  2 1 1 2 2 1 1 2 2 1 1 2
                 . . 1 2 2 1 1 2 2 1 1 2 2 1 . . 
                 . . 1 2 2 1 1 2 2 1 1 2 2 1 . . 
                     2 1 1 2 2 1 1 2 2 1 1 2

    use the          a a b b a a b b a a b b
    dissection       a c c b a c c b a c c b
                     b c c a b c c a b c c a
                     b b a a b b a a b b a a

so the 3*N*Z is tileble.

My Conjectures:
   - the 3*2n*N box is not tileble.
   - the 3*N*N box (3*quadrant) is tileble.

Torsten Sillke