There are two triominoes. One is simple a 1x3 rectangle and the other is a small
L-shaped piece.
Using any combination of these two triominoes and in any orientation, how many ways can a 3 by N rectangle be tiled?
Rotations and reflections of an asymmetrical solution are to be considered distinct.
This problem seems harder than some of the other polyomino packing problems.
The usual recursion strategies will go all the way to term 1, since there are so many building blocks.
I'm going to use this post to share the blocks I've found. If anyone shares a block I've missed, I'll edit this post.
3x1 (no rotations or reflections) no extensions
A
A
A
3x2 (x2 rotation) extends to 3x(3k-1)
AA ...AA
BA B...A
BB BB...
3x3 (no rotations or reflections) no extensions
AAA
BBB
CCC
3x3 (x4 rotation/reflection) extends to 3x(3k)
AAA AAA...
BBC BB...C
BCC B...CC
3x4 (x2 rotation/reflection) extends to 3x(3k+1)
AAAB AAA...B
CCBB CC...BB
CDDD C...DDD
This can now be used to write a sort-of recursion, but since there is a foundational block of every size, it wont just use the previous 2 or 3 terms. It will go all the way down to 1.
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Posted by Jer
on 2023-09-22 08:39:53 |
First thoughts
