Finding all rectangles with fault-free tilings of the P-pentomino
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I am trying to find all rectangles with fault-free tilings of the P pentomino. (A fault is a vertical or horizontal line inside the rectangle that is not crossed by any tiles; a fault-free tiling is thus a tiling without such lines.)
My basic strategy is to find "basic" fault free tilings, and then extend them.
So far, the following extensions are possible:
$5 times 2n$ rectangles can be extended to $5 times (2n + 6k)$ for $n geq 1$.
$m times 2n$ rectangles can be extended to $(m + 5) times 2n$ for $m geq 5$ and $n geq 2$.
$(5m + 5) times (2n + 5)$ rectangles can be extended to $(5m + 15) times (2n + 5)$ for $m, n geq 1$.
And I have the following basic rectangles:
$2 times 5$, $4 times 5$, $6 times 5$
$7 times 10$, $9 times 10$, $11 times 10$
$7 times 15$, $9 times 15$, $11 times 15$, $13 times 15$, $15 times 15$
The idea is to find if all possible rectangles can have fault free tilings, so I use this table for comparison.
My question is, is there some systematic way to deal with this "system of rectangles"? How do I know I have everyone? Or if I can make either list of basic rectangles or the list of extensions shorter?
(The P-pentomino is probably manageable with a bit of care, but I make many mistakes which made me wonder if there is a more systematic way. Also, I plan to examine the Y-pentomino next, and since it has 40 prime rectangles, I expect it to have at least as many basic rectangles.)
This picture summarizes the above. Tilings that only have yellow tiles are "basic"; colored tiles are "cylinders" inserted into basic tilings to extend them.
tiling polyomino
asked Dec 8 '18 at 2:18
Herman TullekenHerman Tulleken
926620
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add a comment |
$begingroup$
I am trying to find all rectangles with fault-free tilings of the P pentomino. (A fault is a vertical or horizontal line inside the rectangle that is not crossed by any tiles; a fault-free tiling is thus a tiling without such lines.)
My basic strategy is to find "basic" fault free tilings, and then extend them.
So far, the following extensions are possible:
$5 times 2n$ rectangles can be extended to $5 times (2n + 6k)$ for $n geq 1$.
$m times 2n$ rectangles can be extended to $(m + 5) times 2n$ for $m geq 5$ and $n geq 2$.
$(5m + 5) times (2n + 5)$ rectangles can be extended to $(5m + 15) times (2n + 5)$ for $m, n geq 1$.
And I have the following basic rectangles:
$2 times 5$, $4 times 5$, $6 times 5$
$7 times 10$, $9 times 10$, $11 times 10$
$7 times 15$, $9 times 15$, $11 times 15$, $13 times 15$, $15 times 15$
The idea is to find if all possible rectangles can have fault free tilings, so I use this table for comparison.
My question is, is there some systematic way to deal with this "system of rectangles"? How do I know I have everyone? Or if I can make either list of basic rectangles or the list of extensions shorter?
(The P-pentomino is probably manageable with a bit of care, but I make many mistakes which made me wonder if there is a more systematic way. Also, I plan to examine the Y-pentomino next, and since it has 40 prime rectangles, I expect it to have at least as many basic rectangles.)
This picture summarizes the above. Tilings that only have yellow tiles are "basic"; colored tiles are "cylinders" inserted into basic tilings to extend them.
tiling polyomino
asked Dec 8 '18 at 2:18
Herman TullekenHerman Tulleken
926620
$endgroup$
add a comment |
$begingroup$
I am trying to find all rectangles with fault-free tilings of the P pentomino. (A fault is a vertical or horizontal line inside the rectangle that is not crossed by any tiles; a fault-free tiling is thus a tiling without such lines.)
My basic strategy is to find "basic" fault free tilings, and then extend them.
So far, the following extensions are possible:
$5 times 2n$ rectangles can be extended to $5 times (2n + 6k)$ for $n geq 1$.
$m times 2n$ rectangles can be extended to $(m + 5) times 2n$ for $m geq 5$ and $n geq 2$.
$(5m + 5) times (2n + 5)$ rectangles can be extended to $(5m + 15) times (2n + 5)$ for $m, n geq 1$.
And I have the following basic rectangles:
$2 times 5$, $4 times 5$, $6 times 5$
$7 times 10$, $9 times 10$, $11 times 10$
$7 times 15$, $9 times 15$, $11 times 15$, $13 times 15$, $15 times 15$
The idea is to find if all possible rectangles can have fault free tilings, so I use this table for comparison.
My question is, is there some systematic way to deal with this "system of rectangles"? How do I know I have everyone? Or if I can make either list of basic rectangles or the list of extensions shorter?
(The P-pentomino is probably manageable with a bit of care, but I make many mistakes which made me wonder if there is a more systematic way. Also, I plan to examine the Y-pentomino next, and since it has 40 prime rectangles, I expect it to have at least as many basic rectangles.)
This picture summarizes the above. Tilings that only have yellow tiles are "basic"; colored tiles are "cylinders" inserted into basic tilings to extend them.
tiling polyomino
asked Dec 8 '18 at 2:18
Herman TullekenHerman Tulleken
926620
$endgroup$
I am trying to find all rectangles with fault-free tilings of the P pentomino. (A fault is a vertical or horizontal line inside the rectangle that is not crossed by any tiles; a fault-free tiling is thus a tiling without such lines.)
My basic strategy is to find "basic" fault free tilings, and then extend them.
So far, the following extensions are possible:
$5 times 2n$ rectangles can be extended to $5 times (2n + 6k)$ for $n geq 1$.
$m times 2n$ rectangles can be extended to $(m + 5) times 2n$ for $m geq 5$ and $n geq 2$.
$(5m + 5) times (2n + 5)$ rectangles can be extended to $(5m + 15) times (2n + 5)$ for $m, n geq 1$.
And I have the following basic rectangles:
$2 times 5$, $4 times 5$, $6 times 5$
$7 times 10$, $9 times 10$, $11 times 10$
$7 times 15$, $9 times 15$, $11 times 15$, $13 times 15$, $15 times 15$
The idea is to find if all possible rectangles can have fault free tilings, so I use this table for comparison.
My question is, is there some systematic way to deal with this "system of rectangles"? How do I know I have everyone? Or if I can make either list of basic rectangles or the list of extensions shorter?
(The P-pentomino is probably manageable with a bit of care, but I make many mistakes which made me wonder if there is a more systematic way. Also, I plan to examine the Y-pentomino next, and since it has 40 prime rectangles, I expect it to have at least as many basic rectangles.)
This picture summarizes the above. Tilings that only have yellow tiles are "basic"; colored tiles are "cylinders" inserted into basic tilings to extend them.
tiling polyomino
tiling polyomino
asked Dec 8 '18 at 2:18
Herman TullekenHerman Tulleken
926620
asked Dec 8 '18 at 2:18
Herman TullekenHerman Tulleken
926620
edited Dec 8 '18 at 2:27
Herman Tulleken
asked Dec 8 '18 at 2:18
Herman TullekenHerman Tulleken
926620
asked Dec 8 '18 at 2:18
Herman TullekenHerman Tulleken
926620
asked Dec 8 '18 at 2:18
Herman TullekenHerman Tulleken
926620
926620
add a comment |
add a comment |
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