 All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars  perplexus dot info  Splitting a 2 by N Rectangle (Posted on 2009-11-11) Find an expression which yields the number of ways to split a 2 by N rectangle into two polyominoes. Rotations and reflections are NOT considered distinct.

For example, if N=1 there is only one way - two 1x1 squares.

If N=2 then there are two ways - one way is a 1x1 square with an "L" shape and two 2x1 rectangles as the other.

If N=3 then there are the 6 ways as depicted:

 a a a a a a
 a a a a a a
 a a a a a a
 a a a a a a
 a a a a a a
 a a a a a a

 See The Solution Submitted by Brian Smith No Rating Comments: ( Back to comment list | You must be logged in to post comments.) re: Table | Comment 2 of 3 | (In reply to Table by brianjn)

I also reached the conclusion that the formula is the triangular numbers, but have been unable to show why.  Specifically I have tried to show that there is a recursive pattern F(N) = F(N-1) + N

The reason why N=2 is one number short is easy to see.   The 2x2 rectangle is the only square, therefore it has extra symmetry.

`##     #+++ and #+`

Are the same dissection only for N=2.

 Posted by Jer on 2009-11-12 10:07:14 Please log in:
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