Single-Digit Patterns

What Is a Conjugate Pair?

A conjugate pair exists when a particular digit has exactly two candidate positions remaining in a single house. Because every house must contain each digit exactly once, exactly one of those two cells must contain the digit. This creates a strong link between the two cells. Strong links are the engine that drives all single-digit pattern techniques. The power comes from connecting two conjugate pairs together through a shared element. Conjugate pairs can occur in any house type: rows, columns, or blocks.

The Shared Structure: Connecting Two Conjugate Pairs

All four single-digit patterns share the same abstract structure: 1. Find two conjugate pairs for the same digit. 2. Connect them through a shared element (a weak link where two cells see each other). 3. Identify the two tails (the non-shared endpoints). 4. At least one tail must contain the digit. 5. Eliminate the digit from any cell that can see both tails. The structure is a 4-node alternating inference chain: strong link, weak link, strong link. The four techniques differ only in which house types provide the links.

Skyscraper

A Skyscraper uses two conjugate pairs that are both in the same type of line (both rows or both columns). The two pairs share one endpoint along a perpendicular line, which provides the weak link. The non-shared endpoints are the tails. At least one must contain the digit. Eliminate from cells seeing both tails. The name comes from the appearance of two parallel "towers" offset by one position, resembling a city skyline.

Two-String Kite

A Two-String Kite uses one row conjugate pair and one column conjugate pair. The two pairs are connected through a block: one endpoint of each pair is in the same block, providing the weak link. The non-shared endpoints are the tails. At least one must contain the digit. Eliminate from cells seeing both tails. The row pair forms one "string" and the column pair forms another, connected at the block where they overlap.

Crane (Turbot Fish)

The Crane connects a block conjugate pair with a line conjugate pair. One endpoint of the block pair shares a row or column with one endpoint of the line pair, providing the weak link. The non-shared endpoints are the tails. At least one must contain the digit. Eliminate from cells seeing both tails. The Crane is sometimes harder to spot because block conjugate pairs are less visually prominent.

Empty Rectangle

The Empty Rectangle replaces one conjugate pair with a pattern within a block. The digit's candidates in the block span exactly two rows and two columns, but one corner of the 2x2 intersection is empty. This creates an effective strong link through the block. Combined with an external conjugate pair that passes through the block, this produces two tails. At least one must contain the digit. Eliminate from cells seeing both. The key to spotting an Empty Rectangle is looking for blocks where a digit's candidates form an L-shape or T-shape.

How the Four Techniques Relate to Each Other

They are four manifestations of the same structure: strong link -- weak link -- strong link, on a single digit. Skyscraper: Row + shared column + row (or column + shared row + column) Two-String Kite: Row + shared block + column Crane: Block + shared line + line Empty Rectangle: ER block pattern + shared line + line A solver could implement all four as a single algorithm. The technique names help human solvers recognize visually distinct patterns. All four sit at the same difficulty level. None is inherently harder; they just look different on the grid.

How to Find Single-Digit Patterns

Step 1: Pick a digit with a moderate number of remaining candidates. Step 2: Identify all conjugate pairs in rows, columns, and blocks. Step 3: Try connecting pairs through shared elements (column, row, or block). Step 4: Check for useful eliminations by finding cells that see both tails. Tips: - Start with rows and columns that have exactly two candidates. - Practice one technique at a time. - Remember all four produce the same result: elimination from cells seeing both tails.

Difficulty and Where They Fit

All four single-digit patterns sit at Level 5 (Challenging). They are harder than subset methods and basic fish, but easier than advanced chains and complex fish. Many puzzles that resist basic techniques can be cracked with a well-placed Skyscraper or Two-String Kite. They provide an excellent introduction to the concept of alternating inference chains.

Summary

The Skyscraper, Two-String Kite, Crane, and Empty Rectangle are four faces of the same coin. They all use conjugate pairs to build short alternating inference chains on a single digit. Connect two strong links through a weak link, identify the two tails, and eliminate the digit from any cell that can see both tails. The difference is purely geometric. Learn one, and you are well on your way to learning all four.