Find all polyomino $P$ such that we can tile $nP$ with $n^2$ copies of $P$ for all $n\in \mathbb{N}$. ($nP$ is a polynomino similar to $P$ with scale factor $n$)
I conjecture that there are only $4$ types of such polynomino:
-The first trivial type is rectangle with integer side length.
-The second type is created by $3$ rectangles with integer side as follow:
The last two type is polyomino similar to one of two polyominoes below:
Is there any polyomino which not in the $4$ types above and satisfies that property? What if we replace polyomino by polyiamond, polyabolo, polycube,...? And as Timothy suggest, what if we allow all but finitely many $n$?
Here some link aboutrelate to this problem, but none has any attempt to solve it:
-http://www.recmath.org/PolyPages/PolyPages/index.htm?RepO.htmMore rep-tile polyominoes
-https://www.wcupa.edu/sciences-mathematics/mathematics/vNitica/webRepTiles.aspxThe same question but with polygonal
I think for too complex polyomino, combine copies of it would make either more complex polyomino or too simple polyomino, so it can't tile itself. So I guess we need some invariants which measure the complexity of polyomino. And $n=2$ may be the most important case.
Find all polyomino $P$ such that we can tile $nP$ with $n^2$ copies of $P$ for all $n\in \mathbb{N}$. ($nP$ is a polynomino similar to $P$ with scale factor $n$)
I conjecture that there are only $4$ types of such polynomino:
-The first trivial type is rectangle with integer side length.
-The second type is created by $3$ rectangles with integer side as follow:
The last two type is polyomino similar to one of two polyominoes below:
Is there any polyomino which not in the $4$ types above and satisfies that property? What if we replace polyomino by polyiamond, polyabolo, polycube,...? And as Timothy suggest, what if we allow all but finitely many $n$?
Here some link about this problem:
-http://www.recmath.org/PolyPages/PolyPages/index.htm?RepO.htm
-https://www.wcupa.edu/sciences-mathematics/mathematics/vNitica/webRepTiles.aspx
I think for too complex polyomino, combine copies of it would make either more complex polyomino or too simple polyomino, so it can't tile itself. So I guess we need some invariants which measure the complexity of polyomino.
Find all polyomino $P$ such that we can tile $nP$ with $n^2$ copies of $P$ for all $n\in \mathbb{N}$. ($nP$ is a polynomino similar to $P$ with scale factor $n$)
I conjecture that there are only $4$ types of such polynomino:
-The first trivial type is rectangle with integer side length.
-The second type is created by $3$ rectangles with integer side as follow:
The last two type is polyomino similar to one of two polyominoes below:
Is there any polyomino which not in the $4$ types above and satisfies that property? What if we replace polyomino by polyiamond, polyabolo, polycube,...? And as Timothy suggest, what if we allow all but finitely many $n$?
Here some link relate to this problem, but none has any attempt to solve it:
-More rep-tile polyominoes
-The same question but with polygonal
I think for too complex polyomino, combine copies of it would make either more complex polyomino or too simple polyomino, so it can't tile itself. So I guess we need some invariants which measure the complexity of polyomino. And $n=2$ may be the most important case.
Find all polyomino $P$ such that we can tile $nP$ with $n^2$ copies of $P$ for all $n\in \mathbb{N}$. ($nP$ is a polynomino similar to $P$ with scale factor $n$)
I conjecture that there are only $4$ types of such polynomino:
-The first trivial type is rectangle with integer side length.
-The second type is created by $3$ rectangles with integer side as follow:
The last two type is polyomino similar to one of two polyominoes below:
Is there any polyomino which not in the $4$ types above and satisfies that property? What if we replace polyomino by polyiamond, polyabolo, polycube,...? I have searched self-tiling, rep-tile and other similar key wordAnd as Timothy suggest, what if we allow all but I couldn't findfinitely many $n$?
Here some link about this problem:
-http://www.recmath.org/PolyPages/PolyPages/index.htm?RepO.htm
-https://www.wcupa.edu/sciences-mathematics/mathematics/vNitica/webRepTiles.aspx
I think for too complex polyomino, combine copies of it would make either more complex polyomino or too simple polyomino, so it can't tile itself. So I guess we need some invariants which measure the complexity of polyomino.
Find all polyomino $P$ such that we can tile $nP$ with $n^2$ copies of $P$ for all $n\in \mathbb{N}$. ($nP$ is a polynomino similar to $P$ with scale factor $n$)
I conjecture that there are only $4$ types of such polynomino:
-The first trivial type is rectangle with integer side length.
-The second type is created by $3$ rectangles with integer side as follow:
The last two type is polyomino similar to one of two polyominoes below:
Is there any polyomino which not in the $4$ types above and satisfies that property? What if we replace polyomino by polyiamond, polyabolo, polycube,...? I have searched self-tiling, rep-tile and other similar key word, but I couldn't find this problem.
I think for too complex polyomino, combine copies of it would make either more complex polyomino or too simple polyomino, so it can't tile itself. So I guess we need some invariants which measure the complexity of polyomino.
Find all polyomino $P$ such that we can tile $nP$ with $n^2$ copies of $P$ for all $n\in \mathbb{N}$. ($nP$ is a polynomino similar to $P$ with scale factor $n$)
I conjecture that there are only $4$ types of such polynomino:
-The first trivial type is rectangle with integer side length.
-The second type is created by $3$ rectangles with integer side as follow:
The last two type is polyomino similar to one of two polyominoes below:
Is there any polyomino which not in the $4$ types above and satisfies that property? What if we replace polyomino by polyiamond, polyabolo, polycube,...? And as Timothy suggest, what if we allow all but finitely many $n$?
Here some link about this problem:
-http://www.recmath.org/PolyPages/PolyPages/index.htm?RepO.htm
-https://www.wcupa.edu/sciences-mathematics/mathematics/vNitica/webRepTiles.aspx
I think for too complex polyomino, combine copies of it would make either more complex polyomino or too simple polyomino, so it can't tile itself. So I guess we need some invariants which measure the complexity of polyomino.
Find all polyomino $P$ such that we can tile $nP$ with $n^2$ copies of $P$ for all $n\in \mathbb{N}$. ($nP$ is a polynomino similar to $P$ with scale factor $n$)
I conjecture that there are only $4$ types of such polynomino. The:
-The first trivial type is rectangle with integer side length, and three of the other types are polyominoes which.
-The second type is created by $3$ rectangles with integer side as follow:
The last two type is polyomino similar to one of the threetwo polyominoes below:
Is
Is there any polyomino which not in the $4$ types above and satisfies that property? What if we replace polyomino by polyiamond, polyabolo, polycube,...? I have searched self-tiling, rep-tile and other similar key word, but I couldn't find this problem.
I think for too complex polyomino, combine copies of it would make either more complex polyomino or too simple polyomino, so it can't tile itself. So I guess we need some invariants which measure the complexity of polyomino.
Find all polyomino $P$ such that we can tile $nP$ with $n^2$ copies of $P$ for all $n\in \mathbb{N}$. ($nP$ is a polynomino similar to $P$ with scale factor $n$)
I conjecture that there are only $4$ types of such polynomino. The trivial type is rectangle with integer side length, and three of the other types are polyominoes which similar to one of the three polyominoes below:
Is there any polyomino which not in the $4$ types above and satisfies that property? What if we replace polyomino by polyiamond, polyabolo, polycube,...? I have searched self-tiling, rep-tile and other similar key word, but I couldn't find this problem.
I think for too complex polyomino, combine copies of it would make either more complex polyomino or too simple polyomino, so it can't tile itself. So I guess we need some invariants which measure the complexity of polyomino.
Find all polyomino $P$ such that we can tile $nP$ with $n^2$ copies of $P$ for all $n\in \mathbb{N}$. ($nP$ is a polynomino similar to $P$ with scale factor $n$)
I conjecture that there are only $4$ types of such polynomino:
-The first trivial type is rectangle with integer side length.
-The second type is created by $3$ rectangles with integer side as follow:
The last two type is polyomino similar to one of two polyominoes below:
Is there any polyomino which not in the $4$ types above and satisfies that property? What if we replace polyomino by polyiamond, polyabolo, polycube,...? I have searched self-tiling, rep-tile and other similar key word, but I couldn't find this problem.
I think for too complex polyomino, combine copies of it would make either more complex polyomino or too simple polyomino, so it can't tile itself. So I guess we need some invariants which measure the complexity of polyomino.