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I've almost finished this hard sudoku, but I cannot finish it without guessing or making multiple blind steps to see if I'm wrong or not. Even https://www.sudoku-solutions.com/ gives up and says "there is no hint available". I've tried to apply as many solving techniques as possible, but to no avail. Can anyone help?

enter image description here

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    $\begingroup$ It is often a mistake, but some sudoku puzzles have multiple solutions... I'll have a look though. $\endgroup$ – Joshua Bizley Jul 31 at 20:42
  • $\begingroup$ Thank you, I've also just found this: puzzling.stackexchange.com/questions/252/…, from which I understood it's acceptable to make multiple assumptions on multiple cells (what I called "blind steps forward") and see if you come to a dead end or not... $\endgroup$ – Andrejs Jul 31 at 20:50
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    $\begingroup$ It's not enough to see if you come to a dead end though; you'll need to see if the solution you found was unique; otherwise you'll end up claiming to have solved a sudoku that wasn't actually a sudoku. $\endgroup$ – Bass Jul 31 at 20:58
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    $\begingroup$ There is the following nice trick: If every open cell in the sudoku has exactly two possibilities left except for one cell which has three, this cell has one possible number that is a possibility an odd number of times. This is the correct digit for that cell. In this puzzle, 2 is still possible in eleven cells, 4 is possible in six and 8 is possible in 10. Hence, E7 has to be a 2. $\endgroup$ – Eike Schulte Aug 1 at 13:16
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    $\begingroup$ Look up the solving technique called Sashimi-X-Wing. If F1 does not contain a 2, then you have a basic X-Wing on E1, G1, E9 and G9 which eliminates the 2 in E2. Of course if F1 is a 2, then the 2 in E2 is also eliminated. So in no case can E2 contain a 2. $\endgroup$ – Chris Dunaway Aug 1 at 14:13
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If F1 is a 2, then

G1 is "not 2", G9 must be 2, and E9 cannot be a 2.

On the other hand, if F1 is not a 2, then

E2 must be a 2, so E9 cannot be 2.

Therefore,

E9 cannot be 2.

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  • $\begingroup$ Genius, thank you. Is there a name for this logic, such as swordfish? Or just logic? :) $\endgroup$ – Andrejs Jul 31 at 21:27
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    $\begingroup$ @Andrejs sudokusnake.com/coloring.php $\endgroup$ – Andrew Savinykh Aug 1 at 7:39
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    $\begingroup$ @Andrejs I tried to colour them here E9 cannot be be both red and blue, if we consider number 2. $\endgroup$ – Andrew Savinykh Aug 1 at 7:45
  • $\begingroup$ @Andrejs I think it might be called "universal elimination" in formal logic, but sudoku terms are not my strong suit :-) $\endgroup$ – Bass Aug 1 at 7:48
  • $\begingroup$ @Andrejs To add to this answer, look up "Bivalue Universal Grave + 1" or BUG+1. If every cell has only 2 options except for one cell that has 3 options, you call that a BUG+1. $\endgroup$ – Daniel Baliki Aug 1 at 13:26
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Solution method that uses value propagation and digit elimination, but not explicit case-by-case analysis:

G1 = E9. (That's because of how they interact with G9.) That digit is not 2 because that would squeeze out digit 2 from the DEF123 square altogether. Therefore it is 8.

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The answer is

enter image description here

It is the only solution because it follows logically from placing an 8 on A7, and placing a 2 on A7 led to this contradiction for me (no 2 on row 8). Bass provides more intelligent reasoning for this step.

enter image description here

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