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?

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?

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