dobrichev wrote:Maybe close to this subject is this known invalid pattern
from here.
- Code: Select all
....12.34
.134..2..
.423..1..
.21.34...
3..1...42
4..2...13
.34....21
1...234..
2...413..
That's a really cool find!
It also can be proven quite easily.
- Code: Select all
+---------+---------+---------+
| XY . . | . 1 2 | . 3 4 |
| . 1 3 | 4 . . | 2 . . |
| . 4 2 | 3 . . | 1 . . |
+---------+---------+---------+
| . 2 1 | X 3 4 | . . . |
| 3 . . | 1 . . | . 4 2 |
| 4 . . | 2 . . | . 1 3 |
+---------+---------+---------+
| . 3 4 | . . . | Y 2 1 |
| 1 . . | . 2 3 | 4 . . |
| 2 . . | . 4 1 | 3 . . |
+---------+---------+---------+
With r1c1b68:
Xr4c4 => Xr1c1, -Xr7c7
Yr7c7 => Yr1c1
Or with multi-links:
- Code: Select all
+---------+---------+---------+
| * * * | * 1 2 | * 3 4 |
| * 1 3 | 4 . . | 2 . . |
| * 4 2 | 3 . . | 1 . . |
+---------+---------+---------+
| * 2 1 | * 3 4 | * * * |
| 3 . . | 1 . . | * 4 2 |
| 4 . . | 2 . . | * 1 3 |
+---------+---------+---------+
| * 3 4 | * * * | * 2 1 |
| 1 . . | * 2 3 | 4 . . |
| 2 . . | * 4 1 | 3 . . |
+---------+---------+---------+
r147c147b168 / 2 ie. 4-link
21 truths: *-marked cells
20 links: 5 digits with 4 links each
rank -1, ie. contra.
Edit: I've had another look at 6-box rookeries (with three digits on diagonals). We can always morf the puzzle into this state:
- Code: Select all
+---------+---------+---------+
| X . . | X . . | . . . |
| . X . | . X . | . . . |
| . . X | . . X | . . . |
+---------+---------+---------+
| X . . | . . . | X . . |
| . X . | . . . | . X . |
| . . X | . . . | . . X |
+---------+---------+---------+
| . . . | X . . | . . . |
| . . . | . X . | . . . |
| . . . | . . X | . . . |
+---------+---------+---------+
Then by checking the diagonals in b9:
r7c7, r8c8, r9c9;
r7c8, r8c9, r9c7 and
r7c9, r8c7, r9c8
have solutions, there either are three hexagons or there is none, even when morfed there will be even number of diagonals in each direction.
r7c9, r8c8, r9c7;
r7c8, r8c7, r9c9 and
r7c7, r8c9, r9c8
don't have solutions, there is always one hexagon and odd number of diagonals in each direction.
