Skyscraper

Prerequisite: Chain Basics

Description

Skyscraper is a single-digit candidate elimination technique: focus on one digit d.

If digit d appears in exactly two candidate cells in each of two parallel rows/columns (two walls), and one endpoint from each wall lies on the same perpendicular row/column (the base), the other two endpoints are the roofs.
Any candidate d that can see both roofs can be eliminated.

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Explanation

In the image above, the target digit is 9.

You can picture the pattern as a small “building”:

• Two walls (strong links): in column 2, digit 9 appears only in r4c2 and r9c2; in column 8, digit 9 appears only in r6c8 and r9c8
  (exactly two spots in a unit ⇒ a strong link)
• The base (weak link): the two “bottom” endpoints lie on row 9 (the yellow dashed line)
  (a row can’t contain two 9s ⇒ they can’t both be true)
• The roofs: the two top endpoints (the blue cells r4c2 and r6c8)

Now look at the red candidates: candidate 9 in r6c1 and candidate 9 in r4c9. Why can they be removed?

Here’s the short contradiction:

• Assume the red candidate 9 is true (use r6c1 as an example)
• It sees both roofs, so r4c2 ≠ 9 and r6c8 ≠ 9
• Each wall is a strong link, so 9 is forced onto the base endpoints r9c2 and r9c8
• But the base is one row, so it can’t contain two 9s → contradiction

Therefore, the red candidates can be eliminated.

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Examples

The next two images show two different orientations. Use them as reference:

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How to Find a Skyscraper

In a real puzzle, use this checklist:

1. Pick a digit d
2. Find two parallel rows/columns where digit d appears in exactly 2 candidate cells (two walls)
3. Check whether one endpoint from each wall lies on the same perpendicular row/column (the base)
4. The other endpoints are the roofs: candidates that can see both roofs are eliminations