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I would really appreciate some help because I spent more than hours finding another number. It is supposed to be a very hard one but I started well and according to the key, my numbers are correct. Do not mind the grey box, I used some online tool to re-create the puzzle. Thanks in advance enter image description here

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For a logically complete version of Jens's answer (showing that the solution is unique as well as that it exists), try

putting a 9 in the top left corner of the bottom right 3x3 box.

That leads to

enter image description here

which is impossible. So, by contradiction, that cell must be

2, and the complete Sudoku follows.

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In the lower right 3x3 square, try putting a

2 in the top left corner. The rest falls into place.

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  • 2
    $\begingroup$ Thanks but how did you know? Should this not be always deterministic somehow?? $\endgroup$ – John V Jan 3 at 15:51
  • $\begingroup$ Sometimes you need to guess. In this case it was either 2 or 9. My first try worked. $\endgroup$ – Jens Jan 3 at 15:55
  • $\begingroup$ In the bottom squares, the unknown numbers are 29 in both the middle square and the right hand square. In the second row from the top, the unknown numbers are 79 in the same columns as the 29s at the bottom. Therefore you know that guessing a 2 or a 9 in the bottom squares is going to fix the values of 6 squares and eliminate a lot of possibilities elsewhere. So try it and see what happens! $\endgroup$ – alephzero Jan 4 at 0:06
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placing a 2 in the top right 3x3 square, in its top left corner, will lead to a contradiction of two 2's in the same column (the one you placed, and another in the bottom right 3x3 box)

thus

the only number that can go in that specific tile is a 5

the solution then becomes

912|867|534 543|219|678 678|435|921 ---+---+--- 456|723|189 781|954|362 329|186|457 ---+---+--- 867|591|243 295|348|716 134|672|895

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  • $\begingroup$ Sorry I do not follow - there are no 2s so far. I could place the other two (in the bottom square) to the bottom middle box. $\endgroup$ – John V Jan 3 at 18:13
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Look at the middle three squares. The unknown numbers are

.   .   .  |  17  23  237 | 13  .   .

37  .   12 |  .   .   .   | 13  .   27

37  12  .  |  178 238 .   | .   .   27

If you guess any single number in the middle row, that fixes almost everything else in these squares: either

.   .   .  |  1   2   7 | 3   .   .

3   .   2  |  .   .   . | 1   .   7

7   1  .   |  8   3  .  | .   .   2 

or

.   .   .  |  7   23  23  | 1   .   .

7   .   1  |  .   .   .   | 3   .   2

3   2  .   |  1    8  .   | .   .   7

and it soon becomes clear which option solves the whole puzzle.

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