For those, who are not familiar with this symmetry:
You can perform the equivalence operations
- move the bands down (cyclically, i.e. band 1->2, 2->3, 3->1),
- and the stacks left (stack 3->2, 2->1, 1->3).
- Code: Select all
+---------+---------+---------+ +---------+---------+---------+
| . . 1 | . 7 . | . . . | | . . 1 | . 8 . | . . . |
| . . . | 4 x 8 | . 6 . | | . . . | 5 . 9 | . 4 . |
| 7 . . | . . . | 2 . 3 | | 8 . . | . . . | 2 . 3 |
+---------+---------+---------+ +---------+---------+---------+
| . 9 . | . . . | . . 1 | | . 7 . | . . . | . . 1 |
| 6 y 7 | . 5 . | . . . | | 4 . 8 | . 6 . | . . . |
| . . . | 2 . 3 | 9 . . | | . . . | 2 . 3 | 7 . . |
+---------+---------+---------+ +---------+---------+---------+
| . . . | . . 1 | . 8 . | | . . . | . . 1 | . 9 . |
| . 4 . | . . . | 5 z 9 | | . 5 . | . . . | 6 . 7 |
| 2 . 3 | 8 . . | . . . | | 2 . 3 | 9 . . | . . . |
+---------+---------+---------+ +---------+---------+---------+
If then you change the digits after the cycles (465) and (798) (1,2,3 remain the same), you get the original puzzle again here (each 1 keeps its place, each 4 becomes 6 and so on), with the same solution, if the puzzle is unique.
In other words, the boxes change cyclically with the box numbers (168),(249),(357), and the digits with (1),(2),(3),(465),(789), e.g. box 3 with 623, becomes box 5 with 523.
This symmetry is called Jumpimg Diagonals (or Block symmetry).
Then, for any digit in a cell of the solution this transformation must be true too (the transformed puzzle has the same solution), i.e. if you have a digit a in cell x, there must be the transformed digit b in the transformed cell y.
E.g. if r2c5=9, then r5c2 must be 7, and r9c8 must be 8.
1r2c5 implies 1 in r5c2, but this would kill the two possble 1's in box 5.
2r2c5 implies 2 in r5c2, but this would kill all 3 possible 2's in box 1.
