A number game (Sudoku) is a puzzle in which missing numbers are to be filled into a 9 by 9 grid of squares which are subdivided into 3 by 3 boxes
so that every row, every column, and every box contains the numbers 1 through 9.

Methods to solve Sudoku puzzles (or Create Sudoku puzzles) include but not limited to:
OneChoice, Elimination, Subset, Interaction, X-Wing, XY-Wing, Guessing, and Exhaustive Search.

OneChoice
An empty square can only be filled with a number because all other numbers already appear in this square's row, column, or 3 × 3 box.

Elimination
Row-Elimination: In a row, a number $$x$$ can only be filled in one empty square because
all other empty squares in this row cannot be this number.

Column-Elimination: In a column, a number $$x$$ can only be filled in one empty square because
all other empty squares in this column cannot be this number.

Box-Elimination: In a 3 × 3 box, a number $$x$$ can only be filled in one empty square because
all other empty squares in this box cannot be this number.

Interaction
InteractionRowBox: In a row, a number $$x$$ only appears within one 3 × 3 box,
then in this box, number $$x$$ will only be in this row.

InteractionColBox: In a column, a number $$x$$ only appears within one 3× 3 box,
then in this box, number $$x$$ will only be in this column.

InteractionBoxRow: In a 3 × 3 box, a number $$x$$ only appears within one row,
then in this row, number $$x$$ will only be in this box.

InteractionBoxCol: In a 3 × 3 box, a number $$x$$ only appears within one column,
then in this column, number $$x$$ will only be in this box.

Subset
Subset2: One empty square in a row/column/box can only be filled with number $$x$$ or number $$y$$ ($$x \ne y$$),
another empty square in the same row/column/box can also only be filled with number $$x$$ or number $$y$$,
then all other empty squares in this row/column/box can not contain number $$x$$ or number $$y$$. Think why?

SubsetPosition2: In a row/column/box, an untaken number $$x$$ can only be filled in empty square $$A$$ or $$B$$ ,
in the same row/column/box, another untaken number $$y$$ ($$x \ne y$$) can also only be filled in empty square $$A$$ or $$B$$,
then these two empty squares $$A$$ and $$B$$ will be filled with number $$x$$ and number $$y$$ respectively,
or be filled with number $$y$$ and number $$x$$ respectively . Think why?

Subset3: One empty square in a row/column/box can only be filled with number $$x$$ or number $$y$$ or number $$z$$ ($$x \ne y$$) and ($$x \ne z$$) and ($$y \ne z$$),
another empty square in the same row/column/box can also only be filled with number $$x$$ or number $$y$$ or number $$z$$,
a third empty square in the same row/column/box can also only be filled with number $$x$$ or number $$y$$ or number $$z$$,
then all other empty squares in this row/column/box can not contain number $$x$$ or $$y$$ or $$z$$ . Think why?

SubsetPosition3: In a row/column/box, an untaken number $$x$$ can only be filled in empty square $$A$$ or $$B$$ or $$C$$,
in the same row/column/box, another untaken number $$y$$ can also only be filled in empty square $$A$$ or $$B$$ or $$C$$,
in the same row/column/box, a third untaken number $$z$$ can also only be filled in empty square $$A$$ or $$B$$ or $$C$$,
then these three empty squares $$A$$ and $$B$$ and $$C$$ will be filled with numbers $$x, y, z$$, or $$x, z, y$$, or $$y, x, z$$, or $$y, z, x$$, or $$z, x, y$$ or $$z, y, x$$ respectively. Think why?

$$X$$-Wing Series
DoubleRowsOneDigitCheck($$X$$-Wing):
For an untaken number in a row $$O$$, it can only be in either column $$H$$ or Column $$I$$;
For same untaken number in another row $$P$$, it can only be either column $$H$$ or Column $$I$$ also;
Then for column $$H$$ or column $$I$$, this untaken number can only be in row $$O$$ or row $$P$$.
Think why?

DoubleColsOneDigitCheck($$X$$-Wing):
For an untaken number in a column $$H$$, it can only be in either column $$O$$ or Column $$P$$;
For same untaken number in another column $$I$$, it can only be either column $$O$$ or Column $$P$$ also;
Then for column $$O$$ or column $$P$$, this untaken number can only be in row $$H$$ or row $$I$$.
Think why?

TripleRowsOneDigitCheck (including Turbot Fish etc):
For an untaken number in a row $$O$$, it appears in columns $$C_x$$ where there are at least one and at most three columns in $$C_x$$;
For same untaken number in another row $$P$$, it appears in columns $$C_y$$ where there are at least one and at most three columns in $$C_y$$; also;
For same untaken number in a third row $$Q$$, it appears in columns $$C_z$$ where there are at least one and at most three columns in $$C_z$$; also;
The size of the union of $$C_x$$ and $$C_y$$ and $$C_z$$ is three. That is $$|C_x \cup C_y \cup C_z | = 3$$.
Then for the three columns that $$C_x \cup C_y \cup C_z$$, this untaken number can only be in row $$O$$ or row $$P$$ or row $$Q$$.
Think why?

TripleColsOneDigitCheck (including Turbot Fish etc)
For an untaken number in a column $$O$$, it appears in rows $$O$$ where there are at least one and at most three rows in $$R_x$$;
For same untaken number in another column $$P$$, it appears in rows $$P$$ where there are at least one and at most three rows in $$R_y$$; also;
For same untaken number in a third column $$Q$$, it appears in rows $$Q$$ where there are at least one and at most three rows in $$R_z$$; also;
The size of the union of $$R_x$$ and $$R_y$$ and $$R_z$$ is three. That is $$|R_x \cup R_y \cup R_z | = 3$$.
Then for the three rows that $$R_x \cup R_y \cup R_z$$, this untaken number can only be in column $$O$$ or column $$P$$ or column $$Q$$.
Think why?

XY-Wing Series

Two empty cells are common friends if they are in the same row/column/box.
We call two empty cells in the same row/column/box as row/column/box friend.

XY-Wing-RowBox: For an empty cell which can be filled with either number $$x$$ or number $$y$$,
one of its row friend $$A$$ can be filled with either number $$x$$ or number $$z$$,
and one of its box friend $$B$$ can be filled with either number $$y$$ or number $$z$$,
then the common friends of $$A$$ and $$B$$ CAN NOT take number $$x$$ or number $$y$$.
Think why?

XY-Wing-ColBox: For an empty cell which can be filled with either number $$x$$ or number $$y$$,
one of its column friend $$A$$ can be filled with either number $$x$$ or number $$z$$,
and one of its box friend $$B$$ can be filled with either number $$y$$ or number $$z$$,
then the common friends of $$A$$ and $$B$$ CAN NOT take number $$x$$ or number $$y$$.
Think why?

Guessing

For an empty square $$A$$ which can be filled with several numbers,
If we try this empty cell be filled with number $$x$$,
and we use OneChoice/Subset/Interaction/X-Wing/XY-Wing to repeatedly solve this puzzle,
and we find that one or more empty squares cannot be filled with any number, then we can conclude that empty square $$A$$ can't be $$x$$.
Think why?

Exhaustive Search

For all the empty squares, try all possible combinations of numbers to fill in them, until we find one that satisfying the constraint
that every row/column/box contains nine different numbers from 1 to 9.
Think why?