Sudoku | Fluo IT

SkyscraperSTEP 0/4

Introduction

The Skyscraper strategy allows us to draw conclusions from two strong pairs of the same digit. We do not need a perfect rectangle like in X-Wing - all we need is the right arrangement of two candidate pairs.

We look for two parallel rows in which the selected digit can appear in exactly two places.
In both pairs, one of the cells must be in the same column. These two cells form the common base of the Skyscraper.
The remaining two cells are the roofs. We know that at least one of them must contain the selected digit.

Let's now look at a concrete example. We will focus on the digit 4.

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SkyscraperSTEP 1/4

Checking the Constraints

First, let's examine the notes for the digit 4. We are looking for two rows in which the 4 has exactly two possible positions.

In Row 1, the digit 4 can only appear in Row 1, Column 1 or Row 1, Column 6.
In Row 5, the digit 4 can only appear in Row 5, Column 2 or Row 5, Column 6.
We therefore have two strong pairs: in each of these rows, the digit 4 must occupy one of the two indicated cells.

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SkyscraperSTEP 2/4

Finding the Pattern

Now we notice the most important element of the entire pattern: both pairs have one cell in Column 6.

The cells Row 1, Column 6 and Row 5, Column 6 form the common base of the Skyscraper.
The remaining cells - Row 1, Column 1 and Row 5, Column 2 - are our roofs.
Since Row 1, Column 6 and Row 5, Column 6 are in the same column, they cannot both contain the digit 4. This forces a 4 to appear in at least one of the roofs.

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SkyscraperSTEP 3/4

Analysis

We do not need to know which roof actually contains the digit 4. We only need to consider both possible scenarios.

If Row 1, Column 6 = 4, then Row 5, Column 6 cannot be 4. In that case, the strong pair in Row 5 forces Row 5, Column 2 = 4.
If, on the other hand, Row 1, Column 6 ≠ 4, then the strong pair in Row 1 forces Row 1, Column 1 = 4.
In both cases, we arrive at the same conclusion: at least one of the roofs - Row 1, Column 1 or Row 5, Column 2 - must contain the digit 4.

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SkyscraperSTEP 4/4

Making the Eliminations

Since one of the two roofs must contain the digit 4, we can eliminate 4 from every cell that sees both roofs.

The cell Row 2, Column 2 sees the roof Row 1, Column 1 through the same 3×3 box and the roof Row 5, Column 2 through Column 2. Therefore, we eliminate the digit 4 in the cell Row 2, Column 2.
The cell Row 3, Column 2 also sees both roofs - one through the same 3×3 box and the other through Column 2. Here too, we eliminate the digit 4.

This is the essence of the Skyscraper: we do not know which roof contains the digit 4, but we know that one of them must contain it. Therefore, no cell that sees both roofs can also contain 4.

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