Introduction to AIC: What Can Alternating Strong and Weak Links Remove?

Read candidates as nodes, rather than cells

An AIC is a chain of alternating strong and weak links. Each node represents a candidate proposition, such as “r1c1 is 1,” rather than the cell itself. Even 1 and 4 in the same cell are separate nodes.

A strong link means that two candidates cannot both be false. If one is false, the other is true. A weak link means that two candidates cannot both be true. If one is true, the other is false. A strong link is not always defined as “if one is true, the other is false.” The cells and units with only two candidates used in this article have both properties.

Conditions — solid, dashed, solid

• If a cell has exactly two candidates in total, you can draw a strong link between them.
• If a number has exactly two possible cells in a row, column or box, there is also a strong link between those two candidates.
• Different candidates in one cell, or the same number in two cells that see each other, form a weak link. A weak link holds even when there are three or more possible positions.
• The chains in this article start and end with strong links, as in strong—weak—strong—weak—strong. If both endpoints have the same number, remove that number from other cells that see both endpoints.

If the starting candidate is true, one endpoint is already true. If it is false, the first strong link makes the next candidate true, the weak link makes the following candidate false, and so on until the ending candidate is true. Therefore, the two endpoints cannot both be false. The condition for removal is seeing both endpoints, not seeing every intermediate cell.

Example — six candidates linking two 1s

c1c2c3c4c5c6c7c8c9

r1r2r3r4r5r6r7r8r9

14

3

5

7

6

48

1489

2

6

2

8

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7

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1

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45

14

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6

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9

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ABCDEF

Reading the candidate nodes in the diagram from A to F gives the following graph. = represents a solid line (strong link), and — represents a dashed line (weak link).

A r1c1(1) = B r1c1(4) — C r1c7(4) = D r4c7(4) — E r4c8(4) = F r4c8(1)

1. A=B: r1c1 has only two candidates, 1 and 4.
2. B—C: the two 4s in row 1 cannot both be true.
3. C=D: the only positions for 4 in column 7 are r1c7 and r4c7.
4. D—E: the two 4s in row 4 cannot both be true.
5. E=F: r4c8 has only two candidates, 1 and 4.

At least one of the endpoints A and F is 1. r1c8 sees A through row 1 and F through column 8, so remove 1. Candidates 1, 4, 8 and 9 become 4, 8 and 9. This does not establish that r1c1 is 1.

When you cannot use it, and common mistakes

• You join any two candidates in a three-candidate cell with a solid line. If the third candidate is true, both chosen candidates could be false.
• You reverse a dashed link as though it were strong. Knowing one end is false does not make the other true.
• You link different numbers in different cells just because they share a row or column. A weak link across a row or column connects candidates for the same number.
• You read two consecutive solid lines as strong links. This proof requires alternating strong and weak roles. A link with both properties can be used in the weak role.
• You remove a candidate from a cell that sees only one endpoint. This misses the case where the other endpoint is true.

AIC also includes longer chains and other endpoint patterns. This article covers only the introductory form with the same number at both ends. This app's logical solver searches within a limit of 8 links, so finding no step does not mean there are no AICs at all.

Practice — identify the link types and the elimination

c1c2c3c4c5c6c7c8c9

r1r2r3r4r5r6r7r8r9

7

38

9

2

138

6

4

58

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38

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ABCDEF

Connect A r1c2(3), B r1c2(8), C r1c9(8), D r5c9(8), E r5c9(3), F r5c6(3) in order. Should each link play a strong or weak role? What can you remove from r1c6?

Show answer

The sequence is strong—weak—strong—weak—strong. A=B connects candidates 3 and 8 in r1c2, and B—C connects the 8s in row 1. C=D connects the two possible cells for 8 in column 9. D—E connects different candidates in the same cell, so they cannot both be true (this cell has only two candidates, so the link also has the strong property, but here it plays the weak role). E=F connects the two possible cells for 3 in row 5. At least one of the endpoints A and F is 3, so remove 3 from r1c6, which sees A through row 1 and F through column 6. Candidates 1, 3 and 8 leave 1 and 8.

Further reading

The prerequisites are How to Use Sudoku Candidate Notes and Do You Need Trial and Error to Solve Sudoku?. Next, read Learn Sudoku.

The example boards were found using the sudoku-core generator and logical solver, rather than copied from external puzzles. The seeds, difficulty and first target step numbers are recorded in the article metadata. Technique definition reference: HoDoKu Chains and Loops.