There are a number of strategies to solving a Sudoku.
Disclaimer: I haven't read the many strategy guides floating around, because I like inventing my own. Perhaps there are "better" methods than the ones I might suggest.
--- One Basic Method ---
Here is a straightforward method you can use. Repeatedly apply any of these four rules, as long as you can find one that applies:
1. The 3x3 SQUARE RULE: For each 3x3 square, check where each remaining digit can go. If a given digit can be in only one cell (because all other cells in the 3x3 square have that digit elsewhere in their row or column), that cell must hold that digit.
Note: By "remaining digit" I mean any digit not already used in that collection of nine.
2. THE ROW RULE: For each row, check where each remaining digit can go. If a given digit can be in only one cell (because all other cells in the row have that digit elsewhere in their 3x3 square or column), that cell must hold that digit.
3. THE COLUMN RULE: For each column, check where each remaining digit can go. If a given digit can be in only one cell (because all other cells in the column have that digit elsewhere in their 3x3 square or row), that cell must hold that digit.
4. THE LEFTOVER RULE: For each cell, check if all but one digit is ruled out (because every other digit occurs in the same row or same column or same 3x3 square). If so, the cell must hold that digit.
As long as any of these four rules is triggered, you get to fill in a cell and start over, since rules that didn't apply previously may now apply. If you eventually get to a stage where none of these rules applies, you'll need some fancier tricks that involve sets of cells instead of one cell at a time. But the above might be sufficient for puzzles of not-too-high difficulty.
--- Optimization ---
You can make it more efficient (and interesting) by picking the most likely rule rather than blindly churning through the rules in order. Here are three ways to pick which digit to check next or rule to try next:
1. The more you see of a given digit, the more likely it is that you can find where other cells with the same digit occur. So if you see a lot of 3's, apply all four rules looking for cells that will hold other 3's before you try less-seen digits.
2. When you apply a rule that lets you fill in a cell, look for other rules that involve the cell you just filled in.
3. The more digits that are already filled into a given 3x3 square or row or column, the more likely it is that one of these rules will help you fill in another of the cells in the same 3x3 square or row or column. So look for crowded 3x3 squares, crowded rows, or crowded columns first. Apply rules 1 and 4 to crowded 3x3 squares, rules 2 and 4 to crowded rows, and rules 3 and 4 to crowded columns.
When you first start a puzzle, Optimization #1 will usually lead you to a number of cell values. Midway through solving it, when finding the next cell to fill in can get harder, use Optimization #2, because filling in one cell is often the trigger that lets you fill in a related cell. Optimization #3 works anytime. When you get near the end, Rule 4 tends to take over.
Most important: Have fun!
If my tips help someone, I'd love to hear about it, since I spent a lot of time writing this up for you guys.
