I could find 7 basic methods to try and solve Sudoku puzzles. Naked Singles (NS), Hidden Singles (HS), Naked Pairs (NP), Hidden Pairs (HP), Intersection Removal (IR), X-Wing (XW), and Y-Wing (YW).
Before we can check any of these methods, we need to calculate the possible values allowed in all the empty cells. For example, for a selected empty cell there is no 4 found in the row, no 4 found in the column, and no 4 found in the 3 x 3 square, then the empty cell might possibly be 4. We do this check for 1 through 9.
Naked Singles (NS): For a region (row, column and 3 x 3 square), if an empty cell has a single possibility that cell must be that value.
Hidden Singles (HS): Hidden Singles is like NS, except that the single possibility is mixed in with other possible numbers in that cell. To find it, we need to scan the row, column and 3 x 3 square to see if there is another cell with a possibility of the same value. If not, this cell must be that value.
Naked Pairs (NP): If a region (row, column and 3 x 3 square) contains two cells that have the same two possible values, and no other values, we have a Naked Pair. Since one value must go in one cell, and the second value must go in the other, you can remove these two values from all the other cells in the the same region.
Hidden Pairs (HP): Hidden Pairs are NP mixed in (hidden) with other possible values in the two cells. Note, to be a HP, these values will not be found in any other cells in the same region. Once we find the HP, we just convert them to NP.
So if two hidden pairs are found in a region, we can remove all other values in the cells we found the hidden pair in, turning the HP into a NP
Intersection Removal (IR): If a region (row, column and 3 x 3 square) has only two or three cells that contain the same possible value, and these two or three cells intersect with another region, you can remove this value from the intersecting region. Note That it will always be a row or col intersecting with a squ, or a squ intersecting with a row or col.
Look for a single possibility value that occurs exactly twice in two rows. If the cells containing the value line up in two columns to form a rectangle, all occurrences of the value may be removed from both columns. This also works the other way, starting with columns and eliminating
Y-wings work by eliminating possibilities from intersections of influence. Each cell exerts a range of influence on all the others cells in the same row, column, and box. Y-wing is a complex, advanced technique, so we present an example:
Suppose a cell has exactly two possibilities A and B. This cell AB is the pivot. Consider two cells, AC and BC, which are influenced by AB, but do not influence each other. If they each share exactly one possibility with AB and exactly one possibility with each other, then the possibility held in common between AC and BC, C, can be eliminated from every cell in the intersection of AC and BC’s ranges of influence.
Note, any time we set a cell, we remove that possibility from all the cells in the affected column, row and square regions.