The left-side grid above is an unsolved puzzle. The grid on the right is the solution to the same puzzle. The rules are very simple. Solving these often isn't.
Each square in the completed puzzle must contain one digit such that each digit used appears exactly once in each row, exactly once in each column, and exactly once in each thicker-bounded region (I've colored them in a checkered pattern to make them stand out more, so you can see what I mean). By convention, the digits from 1 to whatever the side length of the grid is are used, but of course it doesn't really matter. Since 4 wasn't among the givens, I could have used zeroes instead in my answer. But I digress.
This sample is practically trivial to solve, but I'll provide a walkthrough so you can see how I'd approach it. First, there needs to be a 1 in the upper-right region (yellowish), and it can't be in the second row (from the top), since there's a 1 in that row already - so it must be in the first row, to the left of the 3. Looking down, there's a 2 in the third column (from the left), so the 2 in the upper-right region must be below the 3. That leaves the 4 to fill in the last box in that region. Now the second row and third column each have only a single empty cell; both are missing the 3. Adding those in, the upper-left region (bluish) needs 2 and 4, and with the given 2 in the first column, the 2 in the upper-left region will have to be in the second column, placing the 4 in the first. Now the first column has only a single empty cell left, which must be a 1. To finish, note that three each of 1 and 3 have been placed in the puzzle - there's only one of each left. They must go in the rows and columns that haven't gotten them yet, so the cell in the third row and second column must be a 3 and the bottom-right corner a 1. The remaining two cells each get a 4, and the puzzle's finished.
That's easy, right? Indeed it is. The one below, however, is a bloody nuisance. Consider it a gift from me to you. Hopefully, you'll figure out some tricks as you go along. You may need to. I'm feeling generous: you may email me if you get stuck, and I'll give you a hint - but only one, and you have to give me what you've solved so far. In creating the puzzle, I also noted the order in which digits may be added to the puzzle to solve it, so if you tell me where you are I can tell you where you can go next... or what mistake you've made. My email address is on my User Info page. I like email from strangers. I'm collecting an archive of the Japanese spam I've received♥
I've made these sorts of puzzles before, but I had never tried to build one with symmetrical starting digits before, much like those seen in Nikoli publications or at the World Puzzle Championships. It's a lot more appealing that way, and if built right, it doesn't really sacrifice the difficulty any. The number of starting digits given is a good indicator of how tough one of these will be; my 23-digit brainwracker below is definitely high-end. (Dell publications usually offer 30 or more digits, making for very easy and subsequently rather unfulfilling puzzles - and they rarely make them symmetrical.) As always, feel free to email me if you solve it, no matter how long it took you or how many people you figure solved it already. Of course, you are also welcome to post a comment about the puzzle (or my write-up) here, as long as it isn't a hint or solution. - ZM