Remote Pairs

What Are Remote Pairs in Sudoku?

A Remote Pair pattern is a chain of four or more bivalue cells that all contain the same two candidates. Each consecutive pair shares a house. The chain can zigzag freely through the grid. The power comes from the alternating relationship: values must alternate along the chain, forcing cells into two parity groups with opposite values.

The Parity Principle: Why Remote Pairs Work

Even-position cells all hold one value, odd-position cells hold the other. We do not know which group gets which value, but the two groups are guaranteed to hold opposite values.

How Eliminations Work

Two chain cells with different parity cover both values between them. Any external cell seeing both can have both candidates eliminated. This cross-grid elimination is something no pair technique can achieve.

Minimum Chain Length: Why Four Cells Matter

Two cells: naked pair (already handled). Three cells: no useful cross-grid reach. Four cells: cell 0 and cell 3 have opposite parity and may be in completely different grid areas. This is where Remote Pairs becomes genuinely useful.

How to Find Remote Pairs in Your Puzzles

1. Identify all bivalue cells. 2. Group by candidate pair (need at least 4 with same pair). 3. Build the chain through shared houses. 4. Assign parity. 5. Search for elimination targets seeing opposite-parity cells. Tips: Check block connections. Use pencilmark coloring.

Difficulty Classification

Level 7 (Very Hard). Chain identification is non-trivial. Parity reasoning is abstract. The search space is large. But Remote Pairs appears with reasonable frequency and often produces multiple eliminations.

Relationship to Simple Coloring

Remote Pairs is essentially Simple Coloring applied to bivalue cells sharing the same two candidates. The parity assignment is the coloring assignment. Because every cell is bivalue, the chain constrains two digits simultaneously, enabling both candidates to be eliminated from external cells.

Summary

Remote Pairs exploits parity in chains of bivalue cells sharing the same two candidates. Any external cell seeing two chain members of different parity can have both candidates eliminated. Level 7 (Very Hard), closely related to Simple Coloring.