Logic Problem: Given a sukoku with only one solution, is it always possible to solve without trial and error?

Yes. Here's a simple description of the algorithm. Into each unspecified square, write all nine digits. Then, systematically apply all constraints. For each row, if a 3 appears anywhere in the row as the only possibility, erase all the '3' digits from the unresolved squares. Do the same for each column and for each subsquare. If that doesn't completely resolve the puzzle, repeat. Resolving any square imposes new constraints. If you make a complete pass through all constraints without resolving any square, then the problem was not completely constrained, and there is more than one solution.

No, it is not. But there is no straightforward way to explain this logically. An example of a grid which must be solved using other means is:

1.......2

.9.4...5.

..6...7..

.5.9.3...

....7....

...85..4.

7.....6..

.3...9.8.

..2.....1

i really don't know... but you will always have to erase a number if you have it wrong.... unless you memorize the sudoku lol

if so then we would be perfect

huh