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Forcing Chain

Forcing Chain is the generic term for all types of chains and loops which propagate implications from one cell or candidate to another.

People who are not accustomed to chains may find them intimidating at first. For a beginner, it is difficult to locate them in the grid and the question is often asked: "Where do I start?". The answer to this question is actually quite simple. You start at the end, by defining what it is that you want to accomplish with the chain. Once the purpose of a chain is defined, you select the 2 candidates that will be positioned at the start and end of the chain. Make sure what type of relationship must exist between these candidates to prove your theory. Then you must try to find a path that connects these 2 candidates. When the chain has the intended result, you are successful. Otherwise, try to find another path. If there is none, your theory (the purpose of the chain) may be incorrect. In that case, you need to define a new purpose.

Another method to find Forcing Chains is by drawing a B/B Plot. This diagram can help you locate possible chains and loops in the grid. Some Sudoku helper programs can also assist you in locating Forcing Chains by marking up the grid or showing the available strong links.

To develop a new skill, you need to understand the basics, but you also need to practice. Start by examining and replaying the examples. Use a computer solver to show you where the chains are. Collect some puzzles that require simple chains and try to solve them on your own. Get some help from experienced solvers on one of the Sudoku Forums.

A Forcing Chain is actually a move simulator. Instead of placing a digit in a cell, we use the chain to find out what would happen when we make that move. "Forcing" refers to the fact that placing a digit in one cell can force another cell to another digit. If r1c3 has candidates 1 and 2, placing digit 1 in r1c1 would force r1c3 to digit 2. If r1c3 and r5c3 would be the only candidates for digit 1 in column 3, forcing r1c3 to 2 would force r5c3 to 1. The effects from each cell to the next are the implications. A Forcing Chain is therefore also known as an implication chain.

Notation


Symbol	Description
rncn	Cell name
rncn=X	Cell rncn contains digit X
rncn<>X	Candidate for digit X eliminated from cell rncn
A => B	A implies B

Here is an example of a Forcing Chain in simple notation:

r1c1=1 => r1c3<>1 => r1c3=2 => r5c3=1

When a cell is immediately forced to a single digit, you may omit the elimination node for that cell. This makes the chain more compact without losing clarity. This is the result:

r1c1=1 => r1c3=2 => r5c3=1

Rules

A Forcing Chain must follow a few rules to ensure that it contains no ambiguity.

The Forcing Chain must be a single stream of implications which flows from left to right.
A placement can imply an elimination or force a digit which would be the result of that elimination.
An elimination can only force a digit into a bivalue or bilocal cell.
Each implication must be the direct result of the preceding node. It is not allowed to combine the effects of earlier nodes.
No branching is allowed. The result would be a Forcing Net.

Forcing Chain

Notation

Rules

See Also