Which polyominoes and polycubes can be packed, all copies alike, into rectangles, boxes and hyperboxes, and...

I didn't read the new FAQ so far, but the old one didn't say much about
tilings.  As I have made long computations for over one year now, there
is some progress and some notorious problems.

My (incomplete) list of Polycubes, for which I have computed prime
rectangles, boxes, hyperboxes.  I use the height notation, the
Schroeppel notation has been appended two times in brackets.  (The
pentacube numbering is from E. Kuenzell.)

polyominoes with complete list of prime packings:

  I5  (trivial, as it is harmonious)

  P5  (everybody knows)

  L5  (3*3*Z, 3*3*3*30, 3*3*3*45 are new, 3*3*...*3*15 is impossible)

  Y5

  U5

       1 1   [ 1 1 ]
  61:  2 1   [ 3 1 ]

         1
  71:  2 1 1

       2
  41:  1 1 1

list almost complete:

         1 1     The last open case is: 3*5*n with n = 2 (mod 4) and n >= 14.
  31:  2 1       In this case reflections are allowed.

list incomplete:

  N5   (I didn't try this one so far)
  V5
  T5
  F5
  W5
  U6
  G6

       1
  81:  2 1 1

         2
  82:  1 1 1

  21:  1        (only 4*5*6 found so far. It is unique.)
       1 1 2    (With reflections: 3*4*15, 3*5*6, 3*5*9, 4*4*10)

             1
           1
  glider:  1 1 1


           1
       1 1
  **:  1 1     (no N*N. possible e. g. 5*Z, 3*3*10)


           1
           1
  **:  1 1     (is there a rectangle possible?)

no 3-dim boxes found so far:

  Z5
       2
  37:  1 2   (5*N*N is tilable, all n-dim packings found for n>=4)

         2 2'  [    3 2 ]  (2' : only one cube at the second level)
  35:  1 1     [  1 1   ]

         2 1
  33:  1 1

         1
  51:  1 2 1

not boxable in any dimension:

  X5   (N*N*...*N is not tilable, as the corner can not be filled.)


Open Problems:

1) The last tetracube tiling problem: (2nd. Update)

 ______
|\     \               Is it possible to tile a 3*2n*2m box
| \_____\              with the tetracube 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 than 9.
                 I found no 3*2n*N for n = 1..9 (n=10 is half done).
                 The search tree dies out before reaching 4/3 n (3rd dimension).
                 Mike Beeler confirmed my computations for n = 1..8.

 The 3*4*Z is tileable:
    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


 The 3*N*N is tileable:
  as the 3*(4*Z bent) is tileable.

       . .
       . .
     3 3 3 3
     3 3 3 3
     3 3 3 3
     3 3 3 3 3 3 3 3              Try for yourself!
     3 3 3 3 3 3 3 3 . .
     3 3 3 3 3 3 3 3 . .
     3 3 3 3 3 3 3 3

  My Conjecture:
   - the 3*n*N box (one side open) is not tileable for all n>0.

2) Rectangles with the One-sided Y5, P5, L5.

There are three pentominoes, which tile rectangles and are handed.
These are: L, P, and Y.

Show: the only prime rectangles for the one-sided pentominoes are:

  L:  2*5

  P:  2*5

  Y:  5*10  (it tiles a*N (strip) -> 5|n)
            I tried to classify the N*N tilings, but there are too many
            cases.

The handed L4 is long known (AMM (Sep. 1963) 760 E1543).
If you allow for the bent tromino only the half turn, so you have
    o         o o
  o o   and   o    only, then the prime rectangles are 2*3 and 3*2.
There is no odd rectangle possible.  This follows from Conway's proof
of tiling a triangle with   o
                           o o   =  T2.
Shearing T2 gives the two orientations of the tromino.  This has been
noticed by Noam Elkies.

3) T4 rectangles with one hole

  Is there a rectangle n*m with n*m = 1 (mod 4) which can be tiled with
  T4 leaving one hole.
  My Conjecture: No

4) n-bone conjecture

   The n-bone are n spheres in a row glued together.
   So the 5-bone looks like: ooooo
   John H. Conway solved the 3-bone conjecture:

     It is impossible to tile with 3-bones a triangular arrangement of
     spheres.

   See: W. P. Thurston, Conway's Tiling Groups, AMM 97 (1990) 757-773
   You see easily that with 2-bones there is a tiling of the triangle
   or tetrahedron iff the number of spheres is odd.

   On the right is an           2
   example tiling of           2 1
   a triangle with 2-bones    0 0 1

 My extended conjecture is: (the n-bone conjecture)
   There is no tiling of the d-simplex with n-bones, where d>=2 and
   n>=3.  The simplex had to be finite of course.

5) N-pentaglobe tetrahedron

   Is there an easy proof, that 24           o o
                                        o o o
   do not tile a tetrahedron with 8 spheres on the edge.
   (last year I made a list of possible packings of the tetrahedron
    with polyspheres [written in german].  This tile needed the longest
    time.)
   My program takes 100h on a HP850 to show this.
   Is it possible to tile a tetrahedron with 9 spheres on the edge?

6) Y5 packings with minimal period

  A periodic tiling of 3*3*Z with the Y5 has a length which is a
  multiple of 15.  I constructed a tiling with period 30.  Find one with
  period 15.

-- Torsten Sillke <sieda@HRZ.Uni-Bielefeld.DE>, August 1993

[ Editor: see also geometry/tiling/pentacube.contest. ]