Skyscraper

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A Skyscraper is a two-strong-link chain, a type of AIC (alternating inference chain). Its strong links come from conjugate pairs in two rows or two columns. When drawn on the board, the two links are parallel.

How to spot it

• For each candidate x from 1 to 9, examine its conjugate-pair strong links in rows and columns.
• Find two conjugate pairs in two columns (or two rows), so that their strong links are parallel.
• If one cell from each conjugate pair lies in the same row (or column), candidate x can be eliminated from any cell that sees both of the other two cells.

How it works

• Let the two conjugate-pair strong links for candidate x be [A]━[B] and [C]━[D]. If B and C share a row (or column), they see each other and form the weak link [B]┉[C], producing the two-strong-link chain [A]━[B]┉[C]━[D]. By the rules of an AIC, the end nodes A and D also have a strong relationship: at least one of them must be true.
• If A does not contain x, then B must contain x. This prevents C from containing x, which in turn forces D to contain x. The same reasoning works in reverse: if D does not contain x, then A must contain x.
• Therefore, candidate x can be eliminated from any cell that sees both A and D.

Examples

On the board above, candidate 4 forms two parallel conjugate pairs in columns C2 and C9: it occurs in exactly two cells in each column, producing two strong links. Cells R7C2 and R7C9 share a row and form a weak link, giving a two-strong-link chain with end nodes [R1C2:4] and [R2C9:4]. At least one end node must be true, so candidate 4 can be eliminated from R1C8 and R2C3, both of which see the two end nodes.

This is an idealized Skyscraper shape. If the horizontal weak link is viewed as the horizon, the two vertical strong links resemble a pair of skyscrapers. The technique only requires the strong links to be parallel, however; they do not have to be upright.

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The board above contains a horizontal Skyscraper for candidate 1. It has parallel strong links between R3C1 and R3C8, and between R5C2 and R5C8. Cells R3C8 and R5C8 share a column and form the weak link, leaving [R3C1:1] and [R5C2:1] as the end nodes. Candidate 1 can therefore be eliminated from R1C2, R2C2, R4C1, and R6C1.

Note that a Skyscraper can often eliminate more than one candidate. Because the two strong links are parallel and lie in rows or columns, the boxes containing the two end nodes may each contain candidates that form weak links with both end nodes. So when you find the basic shape, inspect the boxes containing the end nodes carefully.

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Here are two more Skyscraper examples in other orientations.