Three layered figures with L3               Torsten Sillke, FRA, 3. Oct. 1995

  If a n-omino p is tileble with L3 and I2 the polycube 3*p is
  tileble with the tricube L3.

  example: p = u-pentomino

       A   B                 3   3
  U5 = A B B   -->  3*U5  =  3 3 3  is L3-tileble.

  This is the case as B3=3*L3 and B2=3*I2 are tileble with L3.

  Problem:
     Find a n-omino p which is not L3, I2 tileble but 3*p is L3 tileble.

  One solution for n=10.

    3 3 3
    3 . 3
  3 3 3 3 3

  Problem:
     Are there n-ominoes with this property without holes?

----------------------------------------------------------------------
  Hints for finding a n-ominoes with this property.

  Several reduction are possible.
  So I cound analyze all n-ominoes for n<=8 with hand.

  This reduction means that if the bigger n-omino has 
  the property a smaller n'-omino must have it too.

  Example-reductions:

	 * * A   -->  A   (A is some n-omino)

	 * * A   -->  A   (A is some n-omino)
	 *

  Others are impossible:
                     *
	 * * *     * * *
	   *         *
	   A         A

  The first interesting case is:

      3                                   1       0       1
    3 3                                 0 0     1 0     0 0
  3 3 3   but this is impossible as:  1 0 1   0 0 0   1 0 1

  A further application of this weighting function is:

      3 3 3                               1 0 1     0 0 0     1 0 1
    3 3 3                               0 0 0     1 0 1     0 0 0
  3 3 3   but this is impossible as:  1 0 1     0 0 0     1 0 1

  Further notice:

  3 3 3                               1 0 1     0 0 0     1 0 1
  3 3 3   the 3-cube                  0 0 0     0 1 0     0 0 0
  3 3 3                               1 0 1     0 0 0     1 0 1

  The 3-cube is possible as everybody knows, but the weighting function
  is sharp. So some L3 positions are forbidden.
----------------------------------------------------------------------
  A related trivial problem:

  L3 tiles 2*L3 and 2*I3. So one can look for non L3, I3 tileble
  polyominoes p for which 2*p is L3 tileble. But this is easy as
      *   *
      * * *  is a minimal example.
	*
----------------------------------------------------------------------
  An n-omino p is called minimal if no tiling of 3*p with L3 contains
  a B2 or B3. The following 19-omino has this property.
  A tiling can contain a 3*2 brick of course but in rotated 
  orientation. These 3*2 bricks are marked with X.

  This are all 4 solutions if the X bricks are fixed.

  .  .  .  X  .  .  .         19-omino
  . 67 67  X  .  .  .
  . 67  .  X  .  .  .         Question:
 69 69 64 64 60 56 56           find an example with n minimal.
  .  .  . 64  . 56  .
  .  .  . 61 61 57  .
  .  .  . 61  .  .  .      1

  .  .  .  X  .  .  .
  . 65 65  X  .  .  .
  . 66  .  X  .  .  .
 69 66 66 60 60 55 55
  .  .  .  X  . 57  .
  .  .  .  X 59 57  .
  .  .  .  X  .  .  .
  
  .  .  . 62  .  .  .
  . 65 62 62  .  .  .
  . 68  . 63  .  .  .
 68 68 63 63 58 58 55
  .  .  .  X  . 58  .
  .  .  .  X 59 59  .
  .  .  .  X  .  .  .
 

  .  .  .  X  .  .  .
  . 67 67  X  .  .  .
  . 67  .  X  .  .  .
 69 69 64 64 59 59 56
  .  .  . 64  . 59  .
  .  .  . 61 61 57  .
  .  .  . 61  .  .  .      2

  .  .  .  X  .  .  .
  . 65 65  X  .  .  .
  . 66  .  X  .  .  .
 69 66 66 60 60 56 56
  .  .  .  X  . 57  .
  .  .  .  X 58 57  .
  .  .  .  X  .  .  .
  
  .  .  . 62  .  .  .
  . 65 62 62  .  .  .
  . 68  . 63  .  .  .
 68 68 63 63 60 55 55
  .  .  .  X  . 55  .
  .  .  .  X 58 58  .
  .  .  .  X  .  .  .

  .  .  .  X  .  .  .
  . 67 67  X  .  .  .
  . 69  .  X  .  .  .
 69 69 64 64 60 56 56
  .  .  . 64  . 56  .
  .  .  . 61 61 57  .
  .  .  . 61  .  .  .      3
  
  .  .  .  X  .  .  .
  . 65 67  X  .  .  .
  . 66  .  X  .  .  .
 68 66 66 60 60 55 55
  .  .  .  X  . 57  .
  .  .  .  X 59 57  .
  .  .  .  X  .  .  .
  
  .  .  . 62  .  .  .
  . 65 62 62  .  .  .
  . 65  . 63  .  .  .
 68 68 63 63 58 58 55
  .  .  .  X  . 58  .
  .  .  .  X 59 59  .
  .  .  .  X  .  .  .
  

  .  .  .  X  .  .  .
  . 67 67  X  .  .  .
  . 69  .  X  .  .  .
 69 69 64 64 59 59 56
  .  .  . 64  . 59  .
  .  .  . 61 61 57  .
  .  .  . 61  .  .  .      4

  .  .  .  X  .  .  .
  . 65 67  X  .  .  .
  . 66  .  X  .  .  .
 68 66 66 60 60 56 56
  .  .  .  X  . 57  .
  .  .  .  X 58 57  .
  .  .  .  X  .  .  .
  
  .  .  . 62  .  .  .
  . 65 62 62  .  .  .
  . 65  . 63  .  .  .
 68 68 63 63 60 55 55
  .  .  .  X  . 55  .
  .  .  .  X 58 58  .
  .  .  .  X  .  .  .

Non minimal examples are:

  3 3 . 3 . 3 3
  3 3 3 3 3 3 3
  . 3 . 3 . 3 .
  3 3 3 3 3 3 3
  . 3 . 3 . 3 .
  3 3 3 3 3 3 3
  3 3 . 3 . 3 3
  
  . . . 3 . . .
  . 3 3 3 3 3 .
  . 3 . 3 . 3 .
  3 3 3 3 3 3 3
  . 3 . 3 . 3 .
  . 3 3 3 3 3 .
  . . . 3 . . .