SF Sudoku Analytical Research

The Net Cast Wide: Mastering the Sudoku Swordfish Strategy

1. Introduction: The Power of Three

In the hierarchy of Sudoku solving techniques, the Swordfish represents the natural evolution of the X-Wing. While the X-Wing locks a candidate into a \(2 \times 2\) relationship, the Swordfish expands this logic to a \(3 \times 3\) grid. It is a "Size 3 Fish" pattern that is often the key to breaking expert-level puzzles.

Like its predecessor, the Swordfish ignores the standard \(3 \times 3\) boxes and focuses entirely on the geometry of rows and columns. It detects a state where the possible locations for a digit are so entangled across three parallel lines that they force eliminations in the orthogonal direction.

2. The Theoretical Framework

2.1 The Logic of Triple Constraints

A Swordfish occurs when a specific candidate digit (let's call it \(k\)) is restricted to no more than three cells in three specific lines (the "Base Sets"). Critically, these cells must align such that they occupy only three lines in the perpendicular direction (the "Cover Sets").

The Theorem:
If digit \(k\) appears in three Rows (\(R_A, R_B, R_C\)) and within those rows, it is restricted to three Columns (\(C_1, C_2, C_3\)):
Then digit \(k\) must appear exactly once in each of the three intersections. Because the three rows must place their instances of digit \(k\) into these three columns, the columns are effectively "full" regarding digit \(k\) for the specific purpose of satisfying these rows.
Result: We can eliminate digit \(k\) from all other rows that pass through columns \(C_1, C_2,\) and \(C_3\).

3. Visual Analysis: Horizontal Swordfish

In this scenario, we identify three rows where a candidate appears restricted to the same three columns.

Scenario: Candidate 5. Base Sets: Row 2, Row 5, Row 8.

5
5
5
5
5
5
5
5
5
5
Figure 1: Horizontal Swordfish on 5s. Base Rows: 2, 5, 8. Cover Columns: 2, 5, 8.

The Logic Trace:

  1. Row Analysis: Rows 2, 5, and 8 have candidates for 5 restricted to Columns 2, 5, and 8.
  2. The Closed Loop: Although no row has all three positions, collectively these three rows claim these three columns completely.
  3. Elimination: No other row (like Row 3 or Row 6) can place a 5 in Columns 2, 5, or 8. The candidates marked in red are removed.

4. Step-by-Step Detection Guide

Spotting a Swordfish is difficult because you cannot simply scan for a shape. You must scan for a relationship.

  1. Highlighter Mode: Use a digital tool or mental filter to highlight ONLY the candidate digit you are analyzing.
  2. Count Candidates: Look for rows (or columns) that have only 2 or 3 highlighted cells.
  3. The "Three-Line" Match: Find three rows that all confine their candidates to the same three columns (or vice versa).
  4. Execute: If the pattern holds, look up and down those columns and delete any candidates that are not part of your three base rows.

5. Conclusion

The Swordfish is a powerful, if rare, tool in the solver's arsenal. It represents the "Size 3" entry in the Fish family (X-Wing is Size 2, Jellyfish is Size 4). Mastering this technique requires moving beyond local pattern recognition to global pattern recognition.