How many ways can a 3x4 rectangle be cut into two polyominoes by cutting along the grid lines? (Not counting reflections and rotations.)
Examples of valid cuts are shown in the first row and invalid cuts are shown in the second row:

+--+--+--+--+   +--+--+--+--+   +--+--+--+--+   +--+--+--+--+
|     |     |   |           |   |  |        |   |  |        |
+  +  +  +  +   +  +--+  +  +   +  +--+--+  +   +  +  +  +  +
|     |     |   |  |  |     |   |        |  |   |  |        |
+--+--+  +  +   +  +--+  +  +   +  +  +--+  +   +  +  +  +  +
|           |   |           |   |     |     |   |  |        |
+--+--+--+--+   +--+--+--+--+   +--+--+--+--+   +--+--+--+--+

+--+--+--+--+   +--+--+--+--+   +--+--+--+--+   +--+--+--+--+
|     |     |   |           |   |  |        |   |     /     |
+  +  +  +  +   +--+--+  +  +   +  +  +--+  +   +  + /+  +  +
|     |     |   |  |  |     |   |        |  |   |   /       |
+--+--+  +  +   +  +--+  +  +   +  +--+--+  +   +  +  +  +  +
|     |     |   |           |   |     |     |   |  |        |
+--+--+--+--+   +--+--+--+--+   +--+--+--+--+   +--+--+--+--+

(In reply to re(3): Solution missed some by Jer)

One solution is to count the special cases (doughnut shapes and reflectable shapes) separately. Then just separate the counting by where the cut enters and exits the rectangle.

It sounds like everyone below got this solution.

Edited on December 8, 2007, 6:38 pm

Posted by Gamer on 2007-12-08 18:37:09