There are four polyominoes, one of each size 14, in the 4x4 grid below. They have been placed so that none of them are able to slide inside the grid.
+++++
 3 3 
+ ++ + +
 3 2 
++ +++
  2 4 4
+ ++ ++
  4 4 1
+++++
Place five polyominoes, one of each size 15, in a 6x6 grid so that none of them can slide.
Similarily place six polyominoes, one of each size 16, in an 8x8 grid.
Note, the polyominoes can be any valid shape of connected squares for each size. The size 3 piece used may be either the L shape or the straight shape. There are 5 possible shapes for the size 4 piece, 12 for the size 5 piece, and 35 for the size 6 piece.
Extra: Does a solution for the 4 pieces in a 4x4 grid exist for each of the size 4 pieces?
Does a solution for the 5 pieces in a 6x6 grid exist for each of the size 5 pieces?
I have discovered at least 105 (+105 reflections, x4 rotations each) solutions for five polyominoes of size 15, in a 6x6 grid so that none of them can slide. One of these many solutions follows:
422
44
54
555
335
13
A six polyominoes solution for an 8x8 grid follows:
6
6
66
44466
5224
555
335
13
Answers to the Extra(s):
Does a solution for the 4 pieces in a 4x4 grid exist for each of the size 4 pieces?
No. I've found only six solutions (plus their reflections and rotations) for one of the five different possible shapes for the size 4 piece. One being demonstrated in the given example.
33 4 2 4 33 224 No solution was found for
32 44 23 44 314 44 the size 4 pieces:
244 334 3344 334 44 433 4444, 444, or 44
441 2231 441 2231 422 31 4 44
Does a solution for the 5 pieces in a 6x6 grid exist for each of the size 5 pieces?
Though a solution may exist for more than one of each of the 35 size 6 pieces, some of the pieces can not provide a solution, therefore the answer is, again, No.
Edited on April 11, 2008, 3:59 am

Posted by Dej Mar
on 20080410 15:17:59 