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

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

I wonder, had anyone noticed that all the polyominoes have 1 mod 3 squares?

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

It was Brian's hint that helped a lot.  I tried making the polyominoes with the L-triominoes, but I couldn't because one square was left over.  Eventually, I figured out that the square left over was always filled.  That appears to be the rule.

Posted by Tristan on 2006-01-29 23:38:55