Non brute-force Sudoku

Question

The mypuzzle.org site (Difficulty = Hard, Date = 2016-07-12) has the following Sudoku puzzle. I filled in a three easy cells (in square brackets) but then got stuck. Aside from brute force, is there a 'nice' continuation to this particular Sudoku?

It would be reasonable to assume that this is a valid Sudoku, so strategies like unique rectangles may be used.

The next step is to fill in the squares with all valid possibilities. E.g. the digits in each blank spot in the top row (call them A3 and A4) are 1 or 8, and as @Marius commented, G2 is either 1 or 5 (we get this from having 1, 2, 5 as the remaining candidates in the bottom-left 3x3 square; row G already has G7=2, so G2 can only be 1 or 5).

This puzzle seems insufficiently constrained, and even the built-in solver from the "To Solver" button gave up. However, the difficulty level isn't the highest and presumably these puzzles are produced and verified algorithmically - hence this post.

(I don't know the answer; spoiler tags aren't needed in answers.)

 3[4]_ | _ 7 2 | 5 9 6
 _ _[9]| 4 _ _ | _ _ 2
 _ _ 7 | _ _ _ | 3 _ 4
-----------------------
 _ _ _ | _ _ _ | _ 4 _
 _ _ _ | _ _ _ | _ _ _
 _ 9 _ | _ _ _ | _ _ _
-----------------------
 8 _ 4 | _ _ _ | 2 _ _
 9 _ _ | _ _ 7 |[4]_ _ 
 7 3 6 | 2 4 _ | _ _ 9

Answer 1

Unfortunately as others have noted their algorithm is not producing "proper" sudokus (proper means "has a unique solution")

Using brute force we can find $80,485$ solutions (I enumerated all using my home brewed brute force solver which utilises a dancing links implementation of Algorithm X).

Since it is not proper, one cannot really employ a couple of solving strategies that rely on the sudoku being proper (such as "unique rectangles" and "bi-value universal grave"), as these can give one solution where there are many, but even if we do use these strategies we still do not find a single solution.

We can employ some of the advanced strategies:
"almost locked sets";
"quad forcing chains";
"death blossom";
"cell forcing chains"; and
"Bowman's bingo"
to reduce the possible values for cells to:

+----------------------------+----------------------------+----------------------------+
|       3        4       18  |      18        7        2  |       5        9        6  |
|     156     1568        9  |       4    13568    13568  |     178      178        2  |
|    1256    12568        7  |   15689    15689    15689  |       3       18        4  |
+----------------------------+----------------------------+----------------------------+
|    1256    25678    12358  | 1356789  1235689   135689  |   16789        4    13578  |
|   12456    25678    12358  | 1356789  1235689  1345689  |   16789   123578    13578  |
|   12456        9    12358  |  135678   123568   134568  |    1678   123578    13578  |
+----------------------------+----------------------------+----------------------------+
|       8       15        4  |   13569    13569    13569  |       2    13567      137  |
|       9      125      125  |   13568    13568        7  |       4    13568     1358  |
|       7        3        6  |       2        4      158  |      18      158        9  |
+----------------------------+----------------------------+----------------------------+

Which is, in fact, the same sets of possible values my solver ends up with.

You can see the strategies above in action without working through them by hand by inputting the puzzle into the solver at suokuwiki.org

---

As a bonus here is one of the minimal additions of ($3$) clues to make the given puzzle as solved so far proper:

+---------+---------+---------+
| 3  4  · | ·  7  2 | 5  9  6 |
| 1  ·  9 | 4  ·  · | ·  ·  2 | < the 1 in this row
| ·  ·  7 | ·  ·  · | 3  ·  4 |
+---------+---------+---------+
| ·  ·  · | ·  ·  · | ·  4  · |
| ·  ·  · | ·  8  · | ·  5  · | < the 8 and 5 in this row
| ·  9  · | ·  ·  · | ·  ·  · |
+---------+---------+---------+
| 8  ·  4 | ·  ·  · | 2  ·  · |
| 9  ·  · | ·  ·  7 | 4  ·  · |
| 7  3  6 | 2  4  · | ·  ·  9 |
+---------+---------+---------+

This is a relatively easy sudoku to solve.

Answer 2

The website Sudoku Wiki has a good solver which shows which technique it is using for each step.

It goes through some of its "extreme solutions" (cell forcing chains, quad forcing chains) and then is unable to solve it.

The website also has a solution counter (using a brute force algorithm) which says there are over 500 possible solutions to the puzzle.

Here are two solutions, marked in red where they differ.