Chute Remote Pairs

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Chute Remote Pairs is a variant of the W-Wing. Its entire structure is confined to one band or stack, and a strong link between two grouped nodes replaces the conjugate-pair pivot of an ordinary W-Wing.

How to spot it

The description below uses a band. For a stack, simply swap band and stack, as well as rows and columns.

• Find two bivalue cells C1 and C2 in the same band. They contain the same candidate pair {w, z}, but lie in different rows and boxes, so they cannot see each other.
• The two bivalue cells occupy two rows of the band, leaving a third row, Ld.
• Let S1 be the intersection of Ld with the box containing C1, S2 the intersection with the box containing C2, and S3 the intersection with the remaining box. Because S3 cannot see either C1 or C2, it is the disjoint intersection.
• If w does not occur in the disjoint intersection S3—where “occur” covers both candidates and placed digits (given or answer)—then candidate z can be eliminated from any cell that sees both C1 and C2.
• If neither w nor z occurs in S3, both w and z can be eliminated from cells that see C1 and C2.
• If both w and z occur in S3, this technique does not apply.

How it works

Like a W-Wing, Chute Remote Pairs can be expressed as an AIC (alternating inference chain).

Suppose z occurs in S3, but w does not:

• Because w does not occur in the disjoint intersection S3, digit w in row Ld must be placed in either S1 or S2. The grouped nodes S1:w and S2:w therefore form a strong link: [S1:w]━[S2:w]. This grouped strong link is the pivot of the pattern.
• S1 shares a box with C1, so S1:w and C1:w form the weak link [S1:w]┉[C1:w]. Similarly, S2:w and C2:w form the weak link [S2:w]┉[C2:w]. Whichever intersection contains w, the corresponding bivalue cell cannot take w.
• Within each bivalue cell, candidates w and z form a strong link: [C1:w]━[C1:z] and [C2:w]━[C2:z]. The bivalue cell that cannot take w must therefore be z, so at least one of C1 and C2 must contain z.
• Together, these links form the AIC [C1:z]━[C1:w]┉[S1:w]━[S2:w]┉[C2:w]━[C2:z]. At least one of C1:z and C2:z must therefore be true, so candidate z can be eliminated from any cell that sees both C1 and C2.

If neither w nor z occurs in S3, each digit can supply a grouped strong-link pivot, producing two parallel AICs. Both candidates can therefore be eliminated.

If both w and z occur in S3, neither digit supplies the grouped strong-link pivot required in Ld, so the technique does not apply.

Examples

On the board above, bivalue cells R2C4 and R4C5 both contain candidates 1 and 7. They lie in stack 2 but cannot see each other. They occupy columns 4 and 5, and boxes 2 and 5, leaving the intersection of column 6 and box 8—{R7C6, R8C6, R9C6}—as the disjoint intersection. It contains a placed 7, but no 1, so digit 1 in column 6 must occur in the intersection with either box 2 or box 5. In the first case, R2C4 cannot be 1 and must be 7; in the second, the same is true of R4C5. Thus, at least one of the two bivalue cells must contain 7, allowing candidate 7 to be eliminated from any cell that sees both of them.

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On the board above, bivalue cells R1C9 and R2C1 both contain candidates 1 and 2 and lie in band 1. Candidate 1 occurs in the disjoint intersection {R3C4, R3C5, R3C6}, but candidate 2 does not. Candidate 1 can therefore be eliminated from any cell that sees both bivalue cells.

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On the board above, bivalue cells R1C5 and R2C1 both contain candidates 4 and 9 and lie in band 1. Neither 4 nor 9 occurs in the disjoint intersection {R3C7, R3C8, R3C9}, so both candidates can be eliminated from any cell that sees the two bivalue cells.