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How can we defeat these horrible Samurai? Let's take a close look at how our solver works.
This image is taken after the first backtracking. That means that every cell that has a value filled in had either only one possible value, or was guessed at...
But... look at that one cell highlighted in red. Right now it can take on the values 3 and 7... but, 3 can't appear in any other cell within its 3x3 square. So, we (as the humans we are) can immediately deduce that 3 must be the only value that can be in that square; as there's no where else it could possibly go.
Why doesn't our solver see that?
The answer is because we (and when I say 'we' I mean 'I') did something stupid. When we set up the constraints for this problem, we said that the value in a cell cannot be equal to any of the other ~20 cells surrounding it. In so doing, we actually threw out a bit of information: not only are the values not equal, but in a given row, column, or square, the values are 'unique', a fun way of saying 'the values 1-9 appear in every grouping'. This implies that if a value can be generated by only one cell in a group without violating a not equal constraint, then that cell must have that value.
Seems small doesn't it? Let's see what the effect is.
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