Sudoku Solving Techniques

1.1 Scanning

Scanning involves examining rows, columns, and boxes to eliminate possible numbers for unsolved cells. Its the foundation of most Sudoku solving methods and can solve many easy puzzles completely.

How to use scanning:

• Look at each empty cell and determine which numbers are already present in its row, column, and box.
• Eliminate those numbers from the possible candidates for that cell.
• If only one candidate remains, fill the cell with that number.

1.2 Cross-Hatching

Cross-hatching is a method of scanning that focuses on a specific number across rows and columns to identify valid placements.

How to use cross-hatching:

• Pick a number (for example, 5) and look at each 3x3 box.
• Check the rows and columns intersecting that box to see if the number is already placed.
• If the number can only fit in one cell in that box, place it there.

1.3 Naked Singles

A naked single occurs when a cell has only one candidate number after elimination.

Steps:

• Look at the possible candidates for each empty cell.
• If a cell has exactly one candidate, this is the only possible number it can take.
• Fill it in immediately.

1.4 Hidden Singles

Hidden singles happen when a candidate number appears only once in the candidates of a particular row, column, or box.

Example: If within a row, the number 7 appears as a candidate in only one cell, that cell must be 7even if there are other candidates in that cell.

2.1 Naked Pairs (and Triples)

Naked pairs are two cells within a unit (row, column, or box) that share exactly the same two candidate numbers. These candidates can be removed from other cells in that unit.

How this works:

• Identify two cells in a unit that are the only ones to contain the same two candidates (e.g., only candidates 3 and 7).
• Since those numbers must be in those two cells, no other cell in that unit can have those candidates.
• Eliminate those candidates from the other cells in the unit.

Naked triples work similarly with three cells containing exactly three candidates among them.

2.2 Hidden Pairs (and Triples)

A hidden pair occurs when two candidate numbers appear only in exactly two cells within a unit, but those cells have other candidates as well.

Procedure:

• Find two numbers that appear only in two cells of a row, column, or box (even if those cells contain more candidates).
• Eliminate all other candidates from those two cells, as those two numbers must occupy those cells.

This technique is useful for narrowing down candidates where naked pairs do not apply.

2.3 Pointing Pairs (Box-Line Reduction)

When a candidate number is confined to one row or column within a 3x3 box, that candidate can be eliminated from other cells in that row or column outside the box.

Explanation:

• Identify a candidate number that appears only in one row (or column) in a 3x3 box.
• Since it must be placed somewhere in that row (or column) within the box, this candidate cannot appear in the same row or column outside that box.
• Eliminate it from those other cells in the row or column.

2.4 Box-Line Reduction

Closely related to pointing pairs, this method looks at candidates confined to rows or columns intersecting 3x3 boxes and reduces possibilities accordingly.

How to use:

• If, in a row or column, all candidates of a particular number lie within a single box, then that number cannot appear outside those candidate cells within the same box.
• This eliminates candidates in other cells of the box.

3.1 X-Wing

The X-Wing technique is used when a candidate number appears exactly twice in two different rows, and the candidate cells line up perfectly in the same two columns (or vice versa).

How it works:

• Find two rows where a candidate number appears only in two cells each, and these cells are in the same two columns.
• These form the corners of a rectangle.
• Because the candidate must occupy one cell in each row and columns intersection, you can eliminate that candidate from the rest of the two columns outside these rows.

This technique helps reduce candidates and progress solving difficult puzzles.

3.2 Swordfish

Swordfish is an extension of the X-Wing technique but involves three rows and three columns. A candidate appears exactly twice or thrice per row in the same set of three columns (or vice versa).

Steps to employ Swordfish:

• Identify three rows where a candidate number appears only in the same three columns.
• Because the candidate must be in one of these cells in each involved row and column, the candidate can be eliminated from those columns outside the three rows.
• The reverse applies if swapping rows and columns.

Swordfish is harder to spot than X-Wing but powerful in complex grids.

3.3 Coloring

Coloring operates on the principle of the two-color rule to eliminate candidates. It tracks strong inferences about candidate placements across rows, columns, and boxes.

How it works:

• Pick a candidate number and color the cells in which this candidate can appear with two colors (say red and blue), alternating as you follow inferences through the grid.
• If a cell sees both colors, that cell cannot have the candidate and the candidate can be eliminated from that cell.
• Additionally, if a color leads to a contradiction, that entire color group can be eliminated.

This technique requires careful logic and is beneficial for very challenging puzzles.

3.4 XY-Wing

The XY-Wing technique involves three cells, each containing candidates that overlap in a way that allows elimination of candidates from other cells.

Pattern:

• Cell A (the pivot) has candidates (X, Y).
• Cell B shares candidate X with cell A and has candidates (X, Z).
• Cell C shares candidate Y with cell A and has candidates (Y, Z).
• Because of this pattern, candidate Z can be eliminated from any cell that sees both cells B and C.

This technique eliminates candidates in places not obvious by simpler methods.