I once wrote a Sudoku solver using the following algorithm: For each empty field, find which values could possibly be written there (which values are not already written in the same row/column/quadrant). Then it applied the following two rule: (1) If there is only one possible value for a field, write it there. (2) For each row/column/quadrant, for each value: if the value can possibly be written only in one field, write it there. Repeat this in a while loop, finish if you checked the rules and couldn't write anything.

No recursion, constant memory usage.

I assumed that this would not be enough. I was like "first I will code the obvious thing, then try the algorithm on examples from a book I bought, find the examples this cannot solve, and then think about further rules which could solve those examples". But as I tried it on the puzzles in the book, it solved all of them.

Which went against my intuition, as I was pretty sure I sometimes use more complicated rules when solving Sudoku by hand. But it turned out that it is somehow really simply for me to miss an opportunity.

This is of course not a proof that each uniquely solvable Sudoku can be solved by this algorithm, but I think that in practice at least 99% of them can be.

Then I tried an opposite thing, to use this solver to create puzzles. Like, create a full board, then try removing random numbers and check if the solver can solve this; stop when no number can be removed. This actually created pretty hard puzzles; harder than any example in my book.