Using 10 L-shaped tiles that pair to make 1/2×2/5 rectangles and 1/3×3/5 rectangles, that makes 128 tilings (each rectangle can be flipped and the square can be rotated),
╔══╤═════╤══╤═════╗
║OO│OOOOO│OO│OOOOO║
║OO└──┐OO│OO└──┐OO║
║OOOOO│OO│OOOOO│OO║
╟─────┼──┴──┬──┴──╢
║OOOOO│OO×OO│OOOOO║
╟──┐OO├──┐OO├──┐OO║
║OO│OO│OO│OO│OO│OO║
║OO└──┤OO└──┤OO└──╢
║OOOOO│OOOOO│OOOOO║
╚═════╧═════╧═════╝
Also found that 18 L-shaped tiles that pair to make 1×1/9 and 2/3×1/6 rectangles can make 2048 tilings (8 of 9 rectangles can be flipped and the square can be flipped and rotated).
The 10-tile square can be squashed into rectangles and tiled, making 30-tile squares, 50-tile squares, 70-tile squares, etc.
The 18-tile square can be tiled, making 162-tile squares, 450-tile squares, 882-tile squares, etc.
450-tile squares based on the 18-tile squares have tiles that pair to make 1/5×1/45 and 2/15×1/30 rectangles.
450-tile squares based on squashed 10-tile squares could have tiles that pair to make 1/90×2/5 and 1/135×3/5 rectangles, 1/30×2/15 and 1/45×3/15 rectangles (same tiles as those based on the 18-tile
squares), 1/18×2/25 and 1/27×3/25 rectangles, ..., 1/6×2/75 and 1/9×3/75 rectangles, or 1/2×2/225 and 1/3×3/225 rectangles.
Seems like there are some sequences in there, such as
The nth number being the number of different ways 2n congruent tiles can fill a square without the center of the square being on the edge of a tile.
The nth number being the number different tile shapes for which 2n congruent tiles can fill a square without the center of the square being on the edge of a tile.
So blowing this up to use only integers, each tile is a "standard" L shape (the bent triomino), stretched to a 6x5 ratio, and our pairs make 12x15 and 10x18 rectangles, for a 30x30 square. Is that correct?
In fact it can be done with as few as 6 tiles if one takes a trapezoidal tile ABCD with side lengths AB = BC = 1, CD = 2, AD = sqrt(2) and angle B = angle C = 90 degrees. (Two of these tiles can be assembled to make an L, and also to make a 1-by-3 rectangle.)
Can I post my write-up here on Tuesday? I've noticed you no longer post these on Twitter (legal reasons with ABC? I hope not but...), so activity is lower over there.
Using 10 L-shaped tiles that pair to make 1/2×2/5 rectangles and 1/3×3/5 rectangles, that makes 128 tilings (each rectangle can be flipped and the square can be rotated),
╔══╤═════╤══╤═════╗
║OO│OOOOO│OO│OOOOO║
║OO└──┐OO│OO└──┐OO║
║OOOOO│OO│OOOOO│OO║
╟─────┼──┴──┬──┴──╢
║OOOOO│OO×OO│OOOOO║
╟──┐OO├──┐OO├──┐OO║
║OO│OO│OO│OO│OO│OO║
║OO└──┤OO└──┤OO└──╢
║OOOOO│OOOOO│OOOOO║
╚═════╧═════╧═════╝
Also found that 18 L-shaped tiles that pair to make 1×1/9 and 2/3×1/6 rectangles can make 2048 tilings (8 of 9 rectangles can be flipped and the square can be flipped and rotated).
The 10-tile square can be squashed into rectangles and tiled, making 30-tile squares, 50-tile squares, 70-tile squares, etc.
The 18-tile square can be tiled, making 162-tile squares, 450-tile squares, 882-tile squares, etc.
450-tile squares based on the 18-tile squares have tiles that pair to make 1/5×1/45 and 2/15×1/30 rectangles.
450-tile squares based on squashed 10-tile squares could have tiles that pair to make 1/90×2/5 and 1/135×3/5 rectangles, 1/30×2/15 and 1/45×3/15 rectangles (same tiles as those based on the 18-tile
squares), 1/18×2/25 and 1/27×3/25 rectangles, ..., 1/6×2/75 and 1/9×3/75 rectangles, or 1/2×2/225 and 1/3×3/225 rectangles.
Seems like there are some sequences in there, such as
The nth number being the number of different ways 2n congruent tiles can fill a square without the center of the square being on the edge of a tile.
The nth number being the number different tile shapes for which 2n congruent tiles can fill a square without the center of the square being on the edge of a tile.
So blowing this up to use only integers, each tile is a "standard" L shape (the bent triomino), stretched to a 6x5 ratio, and our pairs make 12x15 and 10x18 rectangles, for a 30x30 square. Is that correct?
Specifically like this:
https://www.geogebra.org/calculator/knrun3mw
😍
In fact it can be done with as few as 6 tiles if one takes a trapezoidal tile ABCD with side lengths AB = BC = 1, CD = 2, AD = sqrt(2) and angle B = angle C = 90 degrees. (Two of these tiles can be assembled to make an L, and also to make a 1-by-3 rectangle.)
Can I post my write-up here on Tuesday? I've noticed you no longer post these on Twitter (legal reasons with ABC? I hope not but...), so activity is lower over there.
I think that's fair game.
No legal reason why I'm posting less over on Twitter. I honestly just find myself with less enthusiasm for that platform these days.
That's extremely fair. I feel bad for Twitter. (Not for the current company, the old one).
Someone had started to submit it peloton numbers to OEIS last week when I tried to. I don't know how far along it's gotten.
Ah here, it's still in draft form: https://oeis.org/draft/A365905
Oooooo, exciting!
Sharing my full write-up:
https://drive.google.com/file/d/12Was1kkvsfAEO5-IhnhNGfGfyuDfrzbP/view?usp=sharing
This question was tougher than I thought at first, and then easier than the tougher I had thought.