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.) ...
- Code: Select all
+-------+-------+-------+
| . . . | . . . | 6 . 2 |
| . . 7 | 5 . 3 | 8 . . |
| . 9 2 | . . . | . . . |
+-------+-------+-------+
| . 8 3 | . 7 2 | . 1 . |
| . . . | . 4 . | . . . |
| . 1 . | 8 3 . | 9 2 . |
+-------+-------+-------+
| . . . | . . . | 3 8 . |
| . . 8 | 3 . 1 | 5 . . |
| 5 . 1 | . . . | . . . |
+-------+-------+-------+
+-------------------+-------------------+-------------------+
| 1348 345 45 | 1479 189 4789 | 6 34579 2 |
| 146 46 7 | 5 1269 3 | 8 49 149 |
| 13468 9 2 | 1467 168 4678 | 147 3457 13457 |
+-------------------+-------------------+-------------------+
| 469 8 3 | 69 7 2 | 4 1 456 |
| 2679 2567 569 | 169 4 569 | 7 3567 35678 |
| 467 1 456 | 8 3 56 | 9 2 4567 |
+-------------------+-------------------+-------------------+
| 24679 2467 469 | 24679 2569 45679 | 3 8 14679 |
| 24679 2467 8 | 3 269 1 | 5 4679 4679 |
| 5 23467 1 | 24679 2689 46789 | 247 4679 4679 |
+-------------------+-------------------+-------------------+
Simple moves lead to the first of two UR ...
- Code: Select all
+-------------+-------------+---------------+
| 3 4 5 | 79 1 789 | 6 79 2 |
| 1 6 7 | 5 2 3 | 8 *49 *49 |
| 8 9 2 | 47 6 47 | 1 5 347 |
+-------------+-------------+---------------+
| 69 8 3 | 69 7 2 | 4 1 5 |
| 2 5 69 | 1 4 69 | 7 3 8 |
| 7 1 4 | 8 3 5 | 9 2 6 |
+-------------+-------------+---------------+
| 469 7 69 | 2 5 46 | 3 8 1 |
| 46 2 8 | 3 9 1 | 5 467 47 |
| 5 3 1 | 467 8 467 | 2 *4679 *479 |
+-------------+-------------+---------------+
R2C89, R9C89 form a Type-4 Unique Rectangle on <49>:
R9C8 - can remove <4> from <4679> leaving <679>
R9C9 - can remove <4> from <479> leaving <79>
This leads to a second UR ...
- Code: Select all
+-------------+-------------+-------------+
| 3 4 5 | 79 1 789 | 6 79 2 |
| 1 6 7 | 5 2 3 | 8 49 49 |
| 8 9 2 |*47 6 *47 | 1 5 347 |
+-------------+-------------+-------------+
| 69 8 3 | 69 7 2 | 4 1 5 |
| 2 5 69 | 1 4 69 | 7 3 8 |
| 7 1 4 | 8 3 5 | 9 2 6 |
+-------------+-------------+-------------+
| 469 7 69 | 2 5 46 | 3 8 1 |
| 46 2 8 | 3 9 1 | 5 467 47 |
| 5 3 1 |*467 8 *467 | 2 679 79 |
+-------------+-------------+-------------+
R3C46, R9C46 form a Type-2 Unique Rectangle on <47>:
R7C6 - can remove <6> from <46> leaving <4>
R9C8 - can remove <6> from <679> leaving <79>
But the resulting <4> in R7C6 eliminates the only remaining possible positions for a 4 in R9, producing a grid with no solution ...
- Code: Select all
+-------------+-------------+-------------+
| 3 4 5 | 79 1 789 | 6 79 2 |
| 1 6 7 | 5 2 3 | 8 49 49 |
| 8 9 2 | 47 6 47 | 1 5 347 |
+-------------+-------------+-------------+
| 69 8 3 | 69 7 2 | 4 1 5 |
| 2 5 69 | 1 4 69 | 7 3 8 |
| 7 1 4 | 8 3 5 | 9 2 6 |
+-------------+-------------+-------------+
| 469 7 69 | 2 5 4 | 3 8 1 |
| 46 2 8 | 3 9 1 | 5 467 47 |
| 5 3 1 | 67 8 67 | 2 79 79 |
+-------------+-------------+-------------+
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.