RW wrote:That should be quite obvious. The logic behind uniqueness reductions is [...]
Yes, it *should* be obvious. The example is intended for those for whom it isn't.
r.e.s. wrote:fermat wrote:I have been down that road, here is a link to a 5 solution puzzle. [...]
The problem is, there are 5 solutions. Uniqueness just picks one of them.
Just to dispel any notion that unique-solution strategies can be relied upon to "pick one of them" when there are multiple solutions ...
Here's a puzzle with 4 solutions, but UR methods miss all of them, and instead reduce the candidate grid to one with *no* solutions. (I'm sure someone can find a nicer example, as this is just the first one I ran across.) ...
.....
So the blanket use of unique-solution strategies (i.e. when it isn't known whether there's a unique solution) is a kind of guessing -- it doesn't always lead to one of multiple solutions, even though it may just happen to do so for no legitimate reason.
.------------.------------.------------.
| 3 4 5 | 9 1 8 | 6 7 2 |
| 1 6 7 | 5 2 3 | 8 49 49 |
| 8 9 2 | 47 6 47 | 1 5 3 |
:------------+------------+------------:
| 9 8 3 | 6 7 2 | 4 1 5 |
| 2 5 6 | 1 4 9 | 7 3 8 |
| 7 1 4 | 8 3 5 | 9 2 6 |
:------------+------------+------------:
| 46 7 9 | 2 5 46 | 3 8 1 |
| 46 2 8 | 3 9 1 | 5 46 7 |
| 5 3 1 | 47 8 467| 2 469 49 |
'------------'------------'------------'
RW wrote:r.e.s wrote:Just to dispel any notion that unique-solution strategies can be relied upon to "pick one of them" when there are multiple solutions ...
That should be quite obvious. The logic behind uniqueness reductions is:
- Code: Select all
If cell C=a gives a valid solution, then there will be at least one other valid solution to the puzzle.
If we are sure that the puzzle has only one solution, we may eliminate 'a' from 'C'. If it is possible that the puzzle actually has multiple solutions, then the information is totally useless.
RW
fermat wrote:I respect your judgement but still have reservations about whether using UR can actually make the puzzle worse. Can it?
fermat wrote:r.e.s.
Applying a technique on candidates that are not there will certainly fail.
That is what you did because you refused to simplify.
The second UR was already a destroyed when you acted on the first one. You had already found 2 of 4 solutions.
r.e.s. wrote:I "refused to simplify" (i.e. remove some candidates by simple moves) because the purpose was to show that in a multi-solution puzzle, UR moves *could* be made that will produce a grid with no solution.
+-------+-------+-------+
| 9 . . | 4 . . | 2 . . |
| 6 . . | . 8 . | . . . |
| 2 . 7 | . . 9 | 8 . . |
+-------+-------+-------+
| . . . | . . 7 | . 1 . |
| . 2 . | . 3 . | . 7 . |
| . 5 1 | 2 . . | . . . |
+-------+-------+-------+
| . . 9 | 6 . . | 5 . 8 |
| . . . | . 9 . | . . 1 |
| . . 3 | . . 8 | . . 4 |
+-------+-------+-------+
+----------------+----------------+----------------+
| 9 138 58 | 4 167 356 | 2 35 367 |
| 6 34 45 | 357 8 2 | 1 3459 379 |
| 2 134 7 | 135 16 9 | 8 345 36 |
+----------------+----------------+----------------+
| 3 9 *46 | 8 5 7 |*46 1 2 |
| 48 2 *468 | 9 3 1 |*46 7 5 |
| 7 5 1 | 2 46 46 | 39 8 39 |
+----------------+----------------+----------------+
| 14 47 9 | 6 1247 34 | 5 23 8 |
| 458 4678 2 | 357 9 345 | 37 36 1 |
| 15 67 3 | 157 127 8 | 79 269 4 |
+----------------+----------------+----------------+
+----------------+----------------+----------------+
| 9 138 5 | 4 167 356 | 2 35 367 |
| 6 34 45 | 357 8 2 | 1 3459 379 |
| 2 134 7 | 135 16 9 | 8 345 36 |
+----------------+----------------+----------------+
| 3 9 46 | 8 5 7 | 46 1 2 |
| 4 2 8 | 9 3 1 | 46 7 5 |
| 7 5 1 | 2 46 46 | 39 8 39 |
+----------------+----------------+----------------+
| 14 47 9 | 6 1247 34 | 5 23 8 |
| 458 4678 2 | 357 9 345 | 37 36 1 |
| 15 67 3 | 157 127 8 | 79 269 4 |
+----------------+----------------+----------------+
fermat wrote:Much better example.
UR is a shortcut, for me, I don't know of a case where it is the only solution. ( I am no expert. )
Is it a shortcut, in this case, to show that there is no solution?
At least, no valid Sudoku solution?