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:
The following table considers the number of ways that a polyomino of size 'x' can fit into a 2N array as defined in the problem.
Minor [Two rows are considered]
Poly N Cells in length
size 1 2 3 4 5 6 7 8
1 1 1 2 2 3 3 4 4
2 1 2 3 3 4 4 5
3 2 2 3 3 4 4
4 3 3 4 4 5
5 3 3 4 4
6 4 4 5
7 4 4
8 5
Total 1 2 6 10 15 21 28 36
Interval 1 4 4 5 6 7 8
The first two totals and intervals are bit disconcerting but is
apparent from the 6 onwards we have the triangular series.
At best my expression therefore is:
No. of ways = N *(N +1) / 2 for N > 2
Edit only because of formatting issue of this frame.
Edited on November 12, 2009, 10:13 am
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Posted by brianjn
on 2009-11-11 20:29:08 |
Table
