The squares of the six polyominoes depicted below are either filled or empty. What logic was used to determine whether a square filled?

+--+--+--+--+--+              +--+--+
|##|  |##|  |##|              |##|##|
+--+--+--+--+--+  +--+--+     +--+--+
|  |  |  |  |     |##|##|     |##|##|
+--+--+--+--+  +--+--+--+--+  +--+--+
   |##|  |##|  |##|  |  |##|  |  |  |
   +--+--+--+  +--+--+--+--+  +--+--+
      |  |        |##|  |##|     |##|
      +--+        +--+--+--+--+  +--+
                     |##|  |##|
            +--+--+  +--+--+--+
            |  |##|     |##|
+--+--+  +--+--+--+--+  +--+
|##|##|  |##|  |  |  |
+--+--+  +--+--+--+--+     +--+
|##|##|        |##|##|     |##|
+--+--+--+--+  +--+--+  +--+--+--+
   |  |  |  |  |##|##|  |  |  |  |
   +--+--+--+  +--+--+  +--+--+--+
   |##|  |  |           |##|  |##|
   +--+--+--+           +--+--+--+

(In reply to Solution by Tristan)

First, congratulations Tristan.

Second, to answer your question, yes -- I noticed yesterday that the number of squares in each polyominoe was one more than a multiple of three.  I thought it might be important, but didn't know where to go with it.

Third, I just now learned what an L-triominoe is.  And, I believe you are absolutely right.

The squares that are marked filled in Brian Smiths six polyominoes are squares that can remain uncovered when all squares but one are covered by 3-tile L-triominoes.  It does not appear possible to leave a solitary unfilled square uncovered.

Good work.  Good puzzle.  Good night.

Posted by Mindrod on 2006-01-30 02:29:40