Skyscraper

The Skyscraper is a beautiful pattern that builds on the X-Wing concept. Instead of a perfect rectangle, it uses two parallel lines that share one endpoint, creating an asymmetric elimination pattern.

What is a Skyscraper?

A Skyscraper occurs when:

• A candidate appears exactly twice in each of two rows (or columns)
• One pair of cells shares a column (or row), acting as the "base"
• The other pair doesn't align, creating the "towers"

The two aligned cells form the base of the skyscraper. The two non-aligned cells are the towers that stick up at different heights.

Visual Pattern

Imagine looking at city buildings from the side:

        Col A       Col B       Col C
Row 1:            ┌─────┐
                  │  X  │     tower 1
                  └──┬──┘
                     │
Row 3:    ┌─────┐    │
          │  X  │────┤     base (connected)
          └──┬──┘    │
             │       │
             │    ┌──┴──┐
Row 5:       │    │  X  │     tower 2
             │    └─────┘
             │
          ┌──┴──┐
Row 7:    │  X  │
          └─────┘

Wait, let me simplify:

      Col A   Col B   Col C
Row 1:   X             X
Row 5:   X       X

• Row 1 has candidate in columns A and C
• Row 5 has candidate in columns A and B
• Column A is shared (the "base")
• Columns B and C are the "towers"

The Logic

In the example above, consider where the candidate (say, digit 7) must go:

Row 1 must have a 7:

• Either column A or column C

Row 5 must have a 7:

• Either column A or column B

Case analysis:

If Row 1 has 7 in column A:

• Then Row 5 cannot have 7 in column A
• So Row 5 has 7 in column B

If Row 1 has 7 in column C:

• Row 5 could have 7 in either column A or B
• But one of them will be true

The key insight:

• Either Row 1/Col C has the 7, OR Row 5/Col B has the 7 (or both!)
• At least one of the "tower" cells contains the digit

What can we eliminate? Any cell that "sees" both towers can be eliminated. A cell sees a tower if it shares a row, column, or box with it.

Finding Eliminations

The elimination zones are cells that:

• Share a row with one tower AND a column with the other, OR
• Share a column with one tower AND a row with the other, OR
• Share a box with both towers

Typical elimination points:

• The intersection cell of the two tower columns/rows
• Box overlaps with both towers

Worked Example

Consider digit 3 in a grid:

      1   2   3   4   5   6   7   8   9
    ┌───┬───┬───┬───┬───┬───┬───┬───┬───┐
R2  │   │ 3?│   │   │   │   │   │ 3?│   │
    ├───┼───┼───┼───┼───┼───┼───┼───┼───┤
R6  │   │ 3?│   │   │ 3?│   │   │   │   │
    └───┴───┴───┴───┴───┴───┴───┴───┴───┘

• Row 2: candidate 3 in columns 2 and 8
• Row 6: candidate 3 in columns 2 and 5
• Column 2 is the base (shared)
• Columns 5 and 8 are the towers

The skyscraper tells us:

• Either R2C8 = 3, or R6C5 = 3 (or both)
• At least one tower has the 3

What can be eliminated? Any cell that sees both R2C8 and R6C5:

• Same row as one AND same column as other
• Same box as both (unlikely given spread)

Looking at R6C8: shares row 6 with R6C5, shares column 8 with R2C8. If R6C8 had candidate 3, eliminate it!

Looking at R2C5: shares row 2 with R2C8, shares column 5 with R6C5. If R2C5 had candidate 3, eliminate it!

Column-Based Skyscrapers

The same pattern works on columns:

          Col 3           Col 7
    ┌───────────┐   ┌───────────┐
R1  │     3?    │   │           │
    ├───────────┤   ├───────────┤
R4  │     3?    │   │     3?    │
    ├───────────┤   ├───────────┤
R7  │           │   │     3?    │
    └───────────┘   └───────────┘

• Column 3: candidate in rows 1, 4
• Column 7: candidate in rows 4, 7
• Row 4 is the base
• Rows 1 and 7 are the towers

Eliminate 3 from cells that see both R1C3 and R7C7.

How to Find Skyscrapers

Method 1: X-Wing Hunting

1. Look for near-X-Wings (almost a rectangle)
2. If the pattern is off by one row or column, check for Skyscraper
3. Verify exactly 2 candidates per row/column in the pattern

Method 2: Strong Link Chains

1. Find a digit with strong links (exactly 2 candidates in a row/column)
2. Look for two rows with strong links
3. Check if they share exactly one column
4. The unshared columns form the towers

Method 3: Systematic Search

For each digit:

1. List all rows where it appears exactly twice
2. Find pairs of rows sharing exactly one column
3. Verify the pattern and find elimination targets

Skyscraper vs. X-Wing

The Skyscraper is essentially an X-Wing where one corner has shifted.

Common Patterns

The "Staircase" Skyscraper

Towers are diagonally opposite:

X . . X
. . . .
X . X .

Towers at R1C4 and R3C3.

The "Same-Side" Skyscraper

Towers on the same side of the base:

X . X .
. . . .
X . . X

Towers at R1C3 and R3C4.

The "Extended" Skyscraper

Towers are far apart:

X . . . . . . . X
. . . . . . . . .
. . . . . . . . .
X . . . X . . . .

Large grid coverage, powerful eliminations.

Practice Exercise

Find the Skyscraper and elimination:

Digit 5 positions:

• Row 2: columns 3, 9
• Row 8: columns 3, 6

Common Mistakes

Mistake 1: Wrong base identification The base is the SHARED column (or row), not the longer one.

Mistake 2: Missing elimination targets Check ALL cells that see both towers, including box overlaps.

Mistake 3: Eliminating from the wrong cells You eliminate from cells that see BOTH towers, not from the skyscraper cells themselves.

Mistake 4: Confusing with X-Wing If it forms a perfect rectangle, it's an X-Wing with simpler eliminations.

Quick Reference

Skyscraper pattern:

• 2 rows (or columns) with exactly 2 candidates each
• Sharing exactly 1 column (or row) — the base
• Other column (row) positions are towers

Elimination rule:

• Find cells that "see" both tower cells
• Eliminate the candidate from those cells
• "Sees" = shares row, column, or box

Finding it:

1. Look for strong links (2 candidates per unit)
2. Find two parallel units sharing one cross-unit
3. Identify the towers
4. Find intersection zones

When to look:

• X-Wing search comes up empty
• Expert+ difficulty
• Digits with few remaining placements