How do you figure out sudoku puzzles?

Answer 1

The basic Sudoku solving strategy is to look for cells whose values are forced based on the cells already filled. A cell value is forced when all values but one have already been placed in another cell of the same row, column, or box. A value is also forced when, within some row, column, or box, there is only one cell left in which it can go. These take a bit of practice to spot, especially for rows and columns, but it's worth it. Most easy Sudoku can be solved using these 2 rules.

What about harder puzzles? It turns out that there are ways to reduce possibilities based on patterns of possibilities remaining in the unsolved cells. For example, the "Naked Pair" strategy says that when two cells of some group - a row, column, or box - are reduced to the same two possible values, those values can be removed as possibilities for any *other* cells in that group. (Consider what would happen if one of those values was assigned there.) Repeatedly reducing the set of possibilities will eventually crack the puzzle.

Many ingenious strategies have been developed.

There are some at the related links.