... why this won't work all the time? (It works most of the time, but Wayne Gould has confirmed that it won't work all of the time)

It is my conjecture that this straightforward, brute force approach should work:

Pencilmark every unpopulated small square with every digit that could occur there, all digits except those that occur in the small square's row, column, or 3x3 square. (This is admittedly a tedious process, subject to error, but if you're careful you can do it accurately.)

I submit that, because the puzzle has a unique solution, there now MUST EXIST a row, a column, or a 3x3 square in which some pencilmarked digit occurs only once. That digit must populate that small square. So you put the digit in the small square and then, importantly, remove any pencilmarked versions of that digit from the small square's row, column, and 3x3 square.

On you go, continuing to find unique pencilmarked occurrences within a row, column, or 3x3 square, until the puzzle is solved. It's brute force, but it seems like that it must work.

And it has worked, every time I've tried it until last night.

But now, I've got a puzzle that I think disproves my conjecture. I have filled lots of empty cells, and am now facing a situation in which I do not have a row, a column, or a 3x3 square in which there is a unique pencilmarked occurrence. I can make one guess and solve the puzzle easily using the above method, but you should never have to guess. I have triple-checked everything, and I don't think I've made any pencilmark mistakes.

I'd appreciate any insights from the Sudoku community ...

Kern Parker

Asheville, NC