Use the scanning method Since we know that every row, column, and region can only have one instance of each number, you can logically use the scanning method I illustrated in my article "How To Play Sudoku" to find other cells that contain the same number. I won't repeat that technique here. It is essential to learn this technique because every Sudoku puzzle requires it's use. After you have solved (if possible) for the number(s) that had the most "givens", cycle through all the numbers 1 through 9, solving for each number as many as you can using the scanning method. Repeat again and again until you can no longer find a solution for any cell.
Whenever you solve a square, immediately check and update all candidate lists in the same row, column and box.