All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Shapes
Cutting a Rectangle (Posted on 2007-12-07) Difficulty: 3 of 5
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:
+--+--+--+--+   +--+--+--+--+   +--+--+--+--+   +--+--+--+--+
|     |     |   |           |   |  |        |   |  |        |
+  +  +  +  +   +  +--+  +  +   +  +--+--+  +   +  +  +  +  +
|     |     |   |  |  |     |   |        |  |   |  |        |
+--+--+  +  +   +  +--+  +  +   +  +  +--+  +   +  +  +  +  +
|           |   |           |   |     |     |   |  |        |
+--+--+--+--+   +--+--+--+--+   +--+--+--+--+   +--+--+--+--+

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

  Submitted by Brian Smith    
Rating: 4.0000 (1 votes)
Solution: (Hide)
There are 46 ways:

2 segments: 1 way
+--+--+--+--+
|  |        |
+--+  +  +  +
|           |
+  +  +  +  +
|           |
+--+--+--+--+

3 segments: 6 ways
+--+--+--+--+   +--+--+--+--+   +--+--+--+--+   +--+--+--+--+
|  |        |   |     |     |   |  |        |   |     |     |
+  +  +  +  +   +  +  +  +  +   +  +  +  +  +   +--+--+  +  +
|  |        |   |     |     |   |  |        |   |           |
+  +  +  +  +   +  +  +  +  +   +--+  +  +  +   +  +  +  +  +
|  |        |   |     |     |   |           |   |           |
+--+--+--+--+   +--+--+--+--+   +--+--+--+--+   +--+--+--+--+

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

4 segments: 8 ways
+--+--+--+--+   +--+--+--+--+   +--+--+--+--+   +--+--+--+--+
|     |     |   |        |  |   |           |   |  |     |  |
+  +  +  +  +   +--+--+--+  +   +--+--+--+--+   +  +--+--+  +
|     |     |   |           |   |           |   |           |
+--+--+  +  +   +  +  +  +  +   +  +  +  +  +   +  +  +  +  +
|           |   |           |   |           |   |           |
+--+--+--+--+   +--+--+--+--+   +--+--+--+--+   +--+--+--+--+

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

5 segments: 13 ways
+--+--+--+--+   +--+--+--+--+   +--+--+--+--+   +--+--+--+--+
|           |   |           |   |           |   |           |
+  +--+  +  +   +--+  +  +  +   +--+--+  +  +   +--+--+--+  +
|  |  |     |   |  |        |   |     |     |   |        |  |
+--+  +  +  +   +  +--+--+  +   +  +  +--+  +   +  +  +  +  +
|     |     |   |        |  |   |        |  |   |        |  |
+--+--+--+--+   +--+--+--+--+   +--+--+--+--+   +--+--+--+--+

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

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

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

6 segments: 10 ways
+--+--+--+--+   +--+--+--+--+   +--+--+--+--+   +--+--+--+--+
|        |  |   |        |  |   |        |  |   |     |     |
+--+  +--+  +   +--+  +  +  +   +--+--+  +  +   +  +  +--+  +
|  |  |     |   |  |     |  |   |     |  |  |   |        |  |
+  +--+  +  +   +  +--+--+  +   +  +  +--+  +   +--+--+--+  +
|           |   |           |   |           |   |           |
+--+--+--+--+   +--+--+--+--+   +--+--+--+--+   +--+--+--+--+

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

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

7 segments: 8 ways
+--+--+--+--+   +--+--+--+--+   +--+--+--+--+   +--+--+--+--+
|     |     |   |  |  |     |   |           |   |  |        |
+--+  +--+  +   +  +  +--+  +   +--+--+--+  +   +  +--+--+  +
|  |     |  |   |  |     |  |   |        |  |   |        |  |
+  +--+--+  +   +  +--+--+  +   +--+--+--+  +   +--+--+--+  +
|           |   |           |   |           |   |           |
+--+--+--+--+   +--+--+--+--+   +--+--+--+--+   +--+--+--+--+

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

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Puzzle Thoughts K Sengupta2023-04-24 01:39:24
Solution for General Case (NxM rectangle)FrankM2008-01-10 16:18:32
re(4): Solutions good -- explanationGamer2007-12-08 18:37:09
re(3): Solution missed someJer2007-12-07 16:35:59
re(2): Solution missed someDej Mar2007-12-07 16:33:03
re(2): Solution missed someCharlie2007-12-07 15:59:47
Solutioncomputer solutionCharlie2007-12-07 15:34:08
re: Solution missed someJer2007-12-07 14:02:25
SolutionSolutionDej Mar2007-12-07 11:07:53
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (14)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2024 by Animus Pactum Consulting. All rights reserved. Privacy Information