Unsolved square tiles

This page describes why ten square tile problems, which could not be solved by means of the program, have no solution.

1

The first unsolved square tile consists of the files: 
 10 01 00 00 10 11 10
 00 00 10 01 01 01 11
The program found the following partial patterns when trying to find restrictions on each of the tiles (followed by the tiles that are included and the tiles that are not included in the pattern): 
.....   10 01 00    00 10 11 10
..00.   00 00 10    01 01 01 11
.010.
.000.
.....

.....   01 00 00    10 10 11 10
.000.   00 10 01    00 01 01 11
.010.
.00..
.....

........   01 00 10    10 00 11 10
......0.   00 10 01    00 01 01 11
...001..
..0100..
..0010..
..100...
.0......
........

.0.10011001.0...   01 00 11 10    10 00 10
..0.10011001.0..   00 10 01 11    00 01 01
...0.10011001.0.
It is obvious that third and fourth pattern can be combined to fill an infinite square tiling, where each horizontal line has the pattern described by (00(11|1))*. Once the fourth pattern is used, these are the only patterns that can be possible, thus excluding the first and second pattern. This proves that this particular square tiling does not have an infinite solution using all possible tiles in the set.

2

The second unsolved square tile consists of the files: 
 00 10 01 10 01 11 10
 00 10 10 01 01 01 11
The program found the following partial patterns when trying to find restrictions on each of the tiles: 
.00.   00    10 01 10 01 11 10
.00.   00    10 10 01 01 01 11
.00.

.....   10    00 01 10 01 11 10
..10.   10    00 10 01 01 01 11
.010.
.....

........   01 10    00 10 01 11 10
.10101..   10 01    00 10 01 01 11
..10101.
........

.....   01    00 10 01 10 11 10
.010.   01    00 10 10 01 01 11
.01..
.....

........   01 11 10    00 10 10 01
...101..   10 01 11    00 10 01 01
.101101.
..101...
........
The first pattern implies that any 0000 tile (where abcd is to be read as ab above cd) occurs as part of a vertical bar. This also means that if one places a 0 anywhere besides such a vertical bar, that vertical line will also be filled with 0's. If we combine this with second and fourth pattern, it appears that any 1 placed besides a 0-line, of the left or the right also implies a vertical line filled with only 1 with on both sides only 0's. This excludes the existence of the third and fifth pattern. This proves that this particular square tiling does not have an infinite solution using all possible tiles in the set.

3

The third unsolved square tile consists of the files: 
 00 01 00 11 11 10 01
 00 00 10 10 01 11 11
The program found the following partial patterns when trying to find restrictions on each of the tiles: 
.000...   00    01 00 11 11 10 01
..000..   00    00 10 10 01 11 11
...000.

.001100...   01 00 11 10    00 11 01
..001100..   00 10 01 11    00 10 11
...001100.

.....   11 11 10 01    00 01 00
.111.   10 01 11 11    00 00 10
.101.
.111.
.....
The combination of the first two patterns implies that only diagonal patterns for the form ((0)*0110)* can occur. These exclude the occurences of the last two pattern. This proves that this particular square tiling does not have an infinite solution using all possible tiles in the set.

4

The fourth unsolved square tile consists of the files: 
 00 11 01 11 11 10 01
 00 00 10 10 01 11 11
The program found the following partial patterns when trying to find restrictions on each of the tiles: 
...   00 11    01 11 11 10 01
111   00 00    10 10 01 11 11
000
000

........   01 11 10    00 11 11 01
.....11.   10 01 11    00 00 10 11
...1101.
..1011..
..1101..
.1011...
.11.....
........

.....   11 11 10    00 11 01 01
.111.   10 01 11    00 00 10 11
.101.
.11..
.....

.....   11 10 01    00 11 01 11
..11.   01 11 11    00 00 10 10
.101.
.111.
.....
If one wants to extend the first pattern to the top, this is only possible with a row of copies from the fourth pattern. The again results in a horizontal line with 1's. Note that also the third pattern can be used. This means that the first pattern includes the third and fourth pattern, but excludes the second pattern. This proves that this particular square tiling does not have an infinite solution using all possible tiles in the set.

5

The fifth unsolved square tile consists of the files: 
 00 01 00 01 11 11 10 01
 00 00 10 10 10 01 11 11
The program found the following partial patterns when trying to find restrictions on each of the tiles: 
.000...   00    01 00 01 11 11 10 01
..000..   00    00 10 10 10 01 11 11
...000.

.00110...   01 11 10    00 00 01 11 01
..00110..   00 01 11    00 10 10 10 11
...00110.

......   01 11 10    00 01 00 11 01
..110.   10 01 11    00 00 10 10 11
.1011.
.1101.
.011..
......

.....   11 11 10    00 01 00 01 01
.111.   10 01 11    00 00 10 10 11
.101.
.11..
.....

.....   11 10 01    00 01 00 01 11
..11.   01 11 11    00 00 10 10 10
.101.
.111.
.....
This square tile is much like third square tile. The combination of the first three patterns implies that only diagonal patterns for the form ((0)*011)* can occur. These exclude the occurences of the last two pattern. This proves that this particular square tiling does not have an infinite solution using all possible tiles in the set.

6

The sixth unsolved square tile consists of the files: 
 00 10 01 11 01 11 10 01
 00 10 10 10 01 01 11 11
The program found the following partial patterns when trying to find restrictions on each of the tiles: 
.00.   00    10 01 11 01 11 10 01
.00.   00    10 10 10 01 01 11 11
.00.

....   10    00 01 11 01 11 10 01
.10.   10    00 10 10 01 01 11 11
.10.
.1..

........   01 11 10    00 10 11 01 01
......1.   10 01 11    00 10 10 01 11
....101.
..1011..
..1101..
.101....
.1......
........

.....   11 11    00 10 01 01 10 01
.111.   10 01    00 10 10 01 11 11
.101.
.1...
.....

....   01    00 10 01 11 11 10 01
..1.   01    00 10 10 10 01 11 11
.01.
.01.
....

.....   10 01    00 10 01 11 01 11
...1.   11 11    00 10 10 10 01 01
.101.
.111.
.....
The first pattern implies that any 0000 tile occurs as part of a vertical bar. This also means that if one places a 0 anywhere besides such a vertical bar, that vertical line will also be filled with 0's. If a 1 is placed to the right (or similar to the left) this results in a vertical line with 1's. Besides this line there can be a vertical line with 0's, or a line with 0's en 1's using the second, the fourth, the fifth, and the sixth pattern. But in that case, the next vertical line will be filled with 1's. This exclused the third pattern. This proves that this particular square tiling does not have an infinite solution using all possible tiles in the set.

7

The seventh unsolved square tile consists of the files: 
 00 01 11 00 10 11 01 00 11
 00 00 00 10 10 10 01 11 11
The program found the following partial patterns when trying to find restrictions on each of the tiles: 
.011   01 11 01 11    00 00 10 11 00
.011   01 11 01 11    00 00 10 11 00
.000
....

....   00 10 00 11    00 01 11 11 01
000.   10 10 11 11    00 00 00 10 01
110.
110.

....   00 11 10 11    01 00 01 00 11
.111   00 00 10 10    00 10 01 11 11
.100
.100
(The patterns are a subset of the patterns of tenth square tile). The first two patterns can occur in combination, but not in combination with the third pattern. This proves that this particular square tiling does not have an infinite solution using all possible tiles in the set.

8

The eighth unsolved square tile consists of the files: 
 00 10 01 11 00 01 11 01 00 11
 00 00 00 00 10 10 10 01 11 11
The program found the following partial patterns when trying to find restrictions on each of the tiles: 
.011.   01 01 11    00 10 11 00 01 11 00
.01..   00 01 11    00 00 00 10 10 10 11
.000.
.....

.....   11    00 10 01 00 01 11 01 00 11
.11..   00    00 00 00 10 10 10 01 11 11
.000.
.....

.....   10 00 00    00 01 11 01 11 01 11
0000.   00 10 11    00 00 00 10 10 01 11
1110.
..00.
.....

..0111.   00 10 01 01 00 11    01 11 00 11
..011..   00 00 10 01 11 11    00 00 10 10
0001...
11100..
..000..
.......

.....   00 10 11    01 11 00 01 01 00 11
.11..   00 00 10    00 00 10 10 01 11 11
.100.
.000.
.....
The first and the fourth pattern exclude each other. They both need to be included, but because they both have the same triangle filled, they can be placed in such a manner that they must overlap. This proves that this particular square tiling does not have an infinite solution using all possible tiles in the set.

9

The ninth unsolved square tile consists of the files: 
 00 10 11 00 10 00 01 11 10 11
 00 00 00 10 10 01 01 01 11 11
The program found the following partial patterns when trying to find restrictions on each of the tiles: 
11....   10 11 11    00 00 10 00 01 11 10
111.0.   00 00 11    00 10 10 01 01 01 11
11110.
00000.
......

....   10    00 10 11 00 00 01 11 10 11
..0.   10    00 00 00 10 01 01 01 11 11
.10.
.10.
....

.....   00 00 01    00 10 11 10 11 10 11
.000.   10 01 01    00 00 00 10 01 11 11
.010.
.01..

11..   11 01 11 11    00 10 00 10 00 10
111.   00 01 01 11    00 00 10 10 01 11
001.
.01.

.....   00 00 10    10 11 10 00 01 11 11
..00.   00 10 11    00 00 10 01 01 01 11
.100.
.110.
.....
The first and the fourth pattern exclude each other, for the same reason as the previous square tile. This proves that this particular square tiling does not have an infinite solution using all possible tiles in the set.

10

The tenth unsolved square tile consists of the files: 
 00 10 11 10 00 01 11 00 10 11
 00 00 00 10 01 01 01 11 11 11
The program found the following partial patterns when trying to find restrictions on each of the tiles: 
110.   10 11 10 11    00 00 01 11 00 10
110.   00 00 10 11    00 01 01 01 11 11
000.
....

....   00 01 00 11    00 10 11 10 11 10
.000   01 01 11 11    00 00 00 10 01 11
.011
.011

....   00 11 01 11    10 10 00 00 10 11
111.   00 00 01 01    00 10 01 11 11 11
001.
001.

.100   00 10 00 10    10 11 00 01 11 11
.100   00 10 11 11    00 00 01 01 01 11
.111
....
The first two patterns can be combined and the last two patterns can be combined, but it is not possible combine all four. This proves that this particular square tiling does not have an infinite solution using all possible tiles in the set.