Skip to main content
MathOverflow

Return to Answer

fixed a typo
Source Link
Peter Mueller
Peter Mueller
  • 22.5k
  • 1
  • 75
  • 107

It seems that one can color a 15-15-15-30 trapezoid with the given tiles. Here is a picture (sorry about adjacent figures that are the same color, I used random colors so hopefully there are no ambiguities):

enter image description here

In particular, OP pointed out that these scaled 1-1-1-2 trapezoids can tile any equilateral triangle whose side length is a multiple of three. So the original tile can tile any equilateral triangle whose side length is a multiple of 1545.

I bet we didn't see answers for smaller $n$ due to an Aztec-diamond-like boundary condition with the corners.

It seems that one can color a 15-15-15-30 trapezoid with the given tiles. Here is a picture (sorry about adjacent figures that are the same color, I used random colors so hopefully there are no ambiguities):

enter image description here

In particular, OP pointed out that these scaled 1-1-1-2 trapezoids can tile any equilateral triangle whose side length is a multiple of three. So the original tile can tile any equilateral triangle whose side length is a multiple of 15.

I bet we didn't see answers for smaller $n$ due to an Aztec-diamond-like boundary condition with the corners.

It seems that one can color a 15-15-15-30 trapezoid with the given tiles. Here is a picture (sorry about adjacent figures that are the same color, I used random colors so hopefully there are no ambiguities):

enter image description here

In particular, OP pointed out that these scaled 1-1-1-2 trapezoids can tile any equilateral triangle whose side length is a multiple of three. So the original tile can tile any equilateral triangle whose side length is a multiple of 45.

I bet we didn't see answers for smaller $n$ due to an Aztec-diamond-like boundary condition with the corners.

added 113 characters in body
Source Link
Linus Hamilton
Linus Hamilton
  • 1.9k
  • 13
  • 15

It seems that one can color a 15-15-15-30 trapezoid with the given tiles. Here is a picture (sorry about adjacent figures that are the same color, I used random colors so hopefully there are no ambiguities):

enter image description here

In particular, OP pointed out that these scaled 1-1-1-2 trapezoids can tile any equilateral triangle whose side length is a multiple of three. So the original tile can tile any equilateral triangle whose side length is a multiple of 15.

I bet we didn't see answers for smaller $n$ due to an Aztec-diamond-like boundary condition with the corners.

It seems that one can color a 15-15-15-30 trapezoid with the given tiles. Here is a picture (sorry about adjacent figures that are the same color, I used random colors so hopefully there are no ambiguities):

enter image description here

In particular, OP pointed out that these scaled 1-1-1-2 trapezoids can tile any equilateral triangle whose side length is a multiple of three. So the original tile can tile any equilateral triangle whose side length is a multiple of 15.

It seems that one can color a 15-15-15-30 trapezoid with the given tiles. Here is a picture (sorry about adjacent figures that are the same color, I used random colors so hopefully there are no ambiguities):

enter image description here

In particular, OP pointed out that these scaled 1-1-1-2 trapezoids can tile any equilateral triangle whose side length is a multiple of three. So the original tile can tile any equilateral triangle whose side length is a multiple of 15.

I bet we didn't see answers for smaller $n$ due to an Aztec-diamond-like boundary condition with the corners.

Source Link
Linus Hamilton
Linus Hamilton
  • 1.9k
  • 13
  • 15

It seems that one can color a 15-15-15-30 trapezoid with the given tiles. Here is a picture (sorry about adjacent figures that are the same color, I used random colors so hopefully there are no ambiguities):

enter image description here

In particular, OP pointed out that these scaled 1-1-1-2 trapezoids can tile any equilateral triangle whose side length is a multiple of three. So the original tile can tile any equilateral triangle whose side length is a multiple of 15.