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Rigidly Placed Polyominos (Posted on 2008-04-10) Difficulty: 3 of 5
There are four polyominoes, one of each size 1-4, 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 1-5, in a 6x6 grid so that none of them can slide.
Similarily place six polyominoes, one of each size 1-6, 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?

No Solution Yet Submitted by Brian Smith    
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Solution solution Comment 1 of 1

I have discovered at least 105 (+105 reflections, x4 rotations each) solutions for five polyominoes of size 1-5, 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 2008-04-10 15:17:59

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