perplexus.info :: Shapes : Cutting a Rectangle

Cutting a Rectangle (Posted on 2007-12-07)

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 Sengupta	2023-04-24 01:39:24
Solution for General Case (NxM rectangle)	FrankM	2008-01-10 16:18:32
re(4): Solutions good -- explanation	Gamer	2007-12-08 18:37:09
re(3): Solution missed some	Jer	2007-12-07 16:35:59
re(2): Solution missed some	Dej Mar	2007-12-07 16:33:03
re(2): Solution missed some	Charlie	2007-12-07 15:59:47
computer solution	Charlie	2007-12-07 15:34:08
re: Solution missed some	Jer	2007-12-07 14:02:25
Solution	Dej Mar	2007-12-07 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 (3)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:


perplexus dot info