n-square tilings:              Torsten, Sep., 1996
-----------------

Tiling a n*n-square with like polyomnioes with one hole.
The (n-1) * (n+1)-ominoe packings have been analyzed by
Frits G"obel "A nonexact Packing Problem", CFF 40 (1996) 24.
He gives solutions for n<10 except for n=7. Solution given
by him are marked with [G].

(n-1) * (n+1)-ominoe packings:
------------------------------

n=4:  (complete search)

  x x .  [G]
  x x x

  There are two unconnected solutions:

  2 2 3 3    3 1 . 3
  1 2 3 1    3 2 2 1
  1 1 . 1    2 3 3 1
  2 2 3 3    2 1 1 2


n=5:  trivial solution

  1 1 1 2 2  [G]
  1 1 1 2 2
  3 3 . 2 2
  3 3 4 4 4
  3 3 4 4 4


n=6:  (complete search)

  x . .  [G] the only heptomino 
  x x x
  x x x

  . . . . . x
  x x x . . .
  x x x . . .

  . x . .
  . . . .
  x x . x
  x x . x

  x . . . .
  . . . . .
  x x . . x
  x x . . x

  x . . x
  . . . .
  x . . x
  x x . x
  
  x . . x .
  x . . x .
  . . . . .
  x . . x x

  . x . . .
  x x . . x
  . . . . .
  . . . . .
  x x . . x


n=7:  (partial search), Torsten Sillke, 05.09.96

  No packing is possible unless vertex-connectivity 
  is allowed.

  All octominoes (unconnected) fitting into a
  5*6 or 3*7 rectangle have been looked at.
  
  One example of a vertex-connected octomino.

  x . x . x . x
  x x . x . x .   Coordinates: 000 001 003 005 010 012 014 016

  4 4 1 1 0 0 .
  4 2 1 2 0 2 2
  2 4 2 1 2 0 2
  4 3 1 3 0 3 3
  3 4 3 1 3 0 3
  4 5 1 5 0 5 5
  5 4 5 1 5 0 5

  One example of a octomino (with minimal rectangle 6*6).

  a 1 1 1 1 a 1   Coordinates: 000 001 002 003 005 021 031 051
  a b a a 2 a b
  a b 1 b b a b
  a b 1 . 2 a b
  a b a a 2 a b
  a b 1 b b a b
  2 b 2 2 2 2 b

  --------------------------------------
  Tables:
  --------------------------------------

  some Octominoes (unconnected) with minimal rectangle 6*6.

  000 001 002 003 005 021 031 051  Solutions 16

  --------------------------------------

  Octominoes (unconnected) fitting into 5*6.

  000 001 002 003 004 020 032 050  Solutions 4
  000 001 002 015 021 023 025 035  Solutions 4
  000 001 003 015 022 025 031 045  Solutions 4
  000 001 003 015 025 031 042 045  Solutions 4

  --------------------------------------

  Octominoes (unconnected) fitting into 3*7.

  000 001 003 004 021 022 025 026  Solutions 8
  000 001 003 005 010 012 014 016  Solutions 8
  000 001 003 006 010 012 014 015  Solutions 8
  000 001 003 006 021 022 024 025  Solutions 8
  000 001 004 005 010 012 013 016  Solutions 8
  000 001 004 005 021 022 023 026  Solutions 8
  000 001 004 006 010 012 013 015  Solutions 8
  000 001 005 006 021 022 023 024  Solutions 8
  000 002 003 005 011 012 014 016  Solutions 8
  000 002 003 006 011 012 014 015  Solutions 8
  000 002 004 005 011 012 013 016  Solutions 8
  000 002 004 006 011 012 013 015  Solutions 8


n=8:  (partial search)

  x . . .   [G]
  x x x x
  x x x x

  . x . .      2 solutions
  x x x x
  x x x x


n=9:  trivial solution

  1 1 1 1 1 3 3 4 4  [G]
  1 1 1 1 1 3 3 4 4
  2 2 2 2 2 3 3 4 4
  2 2 2 2 2 3 3 4 4
  5 5 6 6 . 3 3 4 4
  5 5 6 6 7 7 7 7 7
  5 5 6 6 7 7 7 7 7
  5 5 6 6 8 8 8 8 8
  5 5 6 6 8 8 8 8 8


n=10:  (partial search)

  no solution found so far.

  All unconnected 11-omines fiting into 5*7
  with a 2*2 square in one corner have been looked at.

  I guess there is no 11-omino solution (connected).

  There is a near miss:

      x                    x
      x x x x x          x x x x x
  a * x x x x x   +  b * x x x x x

  has a unique solution (a=8, b=1), upto a swap of piece 13.
  (Other a, b combinations are impossible.)

  20 20 15 15 15 15 15 10 10 10
  20 20 15 15 15 15 15 14 10 10
  20 20 15 14 14 14 14 14 10 10
  20 20 20 14 14 14 14 14 10 10
  20 20 16 16 16 12 12 12 10 10
  17 17 16 16 13 13 12 12 11 11
  17 17 16 16 13 13 12 12 11 11
  17 17 16 16 13 13 12 12 11 11
  17 17 16 16 13 13 12 12 11 11
  17 17 17  . 13 13 13 11 11 11
  
- - - - - - - - - - - - - - - - - - - - - 

(n+1) * (n-1)-ominoe packings:
------------------------------
In this case there is always a trivial solution: (n+1) * I_(n-1).

n=6: (complete search)

000 001 002 003 004  Solutions 16
x x x x x
000 001 002 003 010  Solutions 16
x
x x x x
000 001 002 010 011  Solutions 144
x x
x x x
000 001 002 030 031  Solutions 320
x x
. .
. .
x x x
000 001 002 030 032  Solutions 32
x . x
. . .
. . .
x x x
000 001 003 010 011  Solutions 16
x x
x x . x
000 001 004 010 014  Solutions 16
x . . . x
x x . . x
000 001 004 011 014  Solutions 16
. x . . x
x x . . x
000 001 010 024 033  Solutions 8
. . . x
. . . . x
x . . .
x x . .
000 001 020 021 040  Solutions 128
x
.
x x
. .
x x
000 001 020 040 041  Solutions 16
x x
. .
. .
. .
x x x
000 002 004 030 032  Solutions 64
x . x
. . .
. . .
x . x . x