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Chute Remote Pair example with digit 3 eliminations
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Chute Remote Pairs

Chute-Based Contradiction Chains

By Minimal Sudoku TeamLast updated:

Chute Remote Pair is a practical shortcut: find a matching pair, check a small set of chute cells, then remove a candidate from cells that see both ends.

Quick Summary
  • Find: Two cells with the same pair (here, 3/8)
  • Check: Key chute cells block one digit of that pair
  • Eliminate: From every cell that sees both endpoints (one or both digits, depending on what is blocked)
  • Use when: You are comfortable with candidate chains and contradiction proofs

The Concept

A chute is a strip of three boxes: either horizontal (rows 1-3, 4-6, 7-9) or vertical (columns 1-3, 4-6, 7-9).

In this strategy, you start from two matching bi-value cells (same pair, same two candidates). Then you use chute restrictions to decide which digit becomes removable.

📦 Term Check
Bi-value cell: a cell with exactly two candidates.
Chain ends: the two endpoint cells of the pair pattern.
See: share a row, column, or box.

Key Chute Cells

For new readers, this is the most important setup step: identify the key chute cells before doing any eliminations.

🧭 What Are Key Chute Cells?
Key chute cells are the blocker cells in the chute that decide whether one digit of the pair is unavailable. If one pair digit is blocked there, eliminations happen on the other digit.

In this screenshot, key chute cells are R7C4, R7C5, and R7C6 (yellow cells in Box 8). Candidate 3 is blocked there, so eliminations are on 8.

The Golden Rule

🔗 Chute Remote Pair Rule
Let the pair be {X,Y}. If key chute cells block X, then eliminate Y from cells that see both pair endpoints.
Elimination Mapping
Blocked in key cells
Eliminate from cells seeing both endpoints
X
Y
Y
X
X and Y
X and Y (double elimination)

Chute geometry helps you decide which digit is blocked first. That makes the final elimination step much easier.

Worked Example

Digit 3 in a Chute Remote Pair

Chute Remote Pair with 3/8 at R9C3 and R8C7

In this position, digit 3 is not in the key cells of Box 8. The remote pair endpoints are:

It does not matter whether those key cells are given clues or solved values. What matters is only that candidate 3 is unavailable there.

  • R9C3 = (3,8)
  • R8C7 = (3,8)
  • Key chute cells: R7C4, R7C5, R7C6

Follow this order:

  1. Check the key chute cells R7C4/R7C5/R7C6.
  2. Candidate 3 is blocked there.
  3. So the removable digit becomes the paired candidate 8.
  4. Now remove 8 from cells that see both endpoints.
Eliminations in This Position
  • Remove 8 from R8C1.
  • Remove 8 from R9C9.

This is possible because 3 cannot be placed in R7 and in columns 4, 5, and 6, which locks the pair logic through the chute.

Why This Works

🧠 The Missing Logic Step
The key is proving the two endpoints cannot both be 3. Once that is true, at least one endpoint must be 8.
  1. Endpoints are R9C3 and R8C7, both {3,8}.
  2. Key chute cells are R7C4, R7C5, R7C6, and they contain no 3.
  3. So in Box 8, digit 3 must be placed in the lower two rows of that box (here: R8C6 or R9C5).
  4. If R9C3 = 3, then row 9 already has a 3, so R9C5 cannot be 3.
  5. If R8C7 = 3, then row 8 already has a 3, so R8C6 cannot be 3.
  6. If both endpoints were 3, both spots for 3 in Box 8 would be blocked. That is impossible.
  7. Therefore, endpoints cannot both be 3, so at least one endpoint must be 8.
  8. Any cell that sees both endpoints cannot be 8, so remove 8 from R8C1 and R9C9.
Contradiction flow (same example):
Chute Remote Pair contradiction proofEndpoints R9C3 and R8C7 both have 3 and 8. Key chute cells R7C4 to R7C6 block 3. Assuming both endpoints are 3 blocks all remaining 3 spots in box 8, creating contradiction, so at least one endpoint is 8 and 8 is removed from cells that see both endpoints.Endpoint AR9C3 {3,8}Endpoint BR8C7 {3,8}Key chute cells block 3R7C4, R7C5, R7C6 contain no 3Remaining 3 spots in Box 8R8C6 or R9C5Assume both endpoints = 3Then R8C6 and R9C5 are both blockedContradiction resolvedAt least one endpoint must be 8Remove 8 at R8C1 and R9C9
If both endpoints were 3, Box 8 would have no place left for 3. So at least one endpoint is 8, enabling the 8 eliminations.

Takeaway: this is a contradiction chain. If both endpoints are the blocked digit, the grid breaks. So at least one endpoint must be the other digit.

How to Find It

1

Find matching bi-value pairs

Look for the same pair (for example 3/8) in multiple cells.
2

Check chute context

See whether the relevant cells lie in a shared row-chute or column-chute structure.
3

Mark key chute cells

Identify the blocker cells in the chute first. Check which pair digit is blocked there.
4

Identify remote endpoints

Pick the two endpoint cells that carry the remote-pair either/or logic. They must not see each other directly.
5

Eliminate from shared visibility

Eliminate the mapped digit(s) from every cell that sees both endpoints.
Common Error
Do not eliminate unless both endpoints are true chain ends and the target cell really sees both.

Single vs Double Eliminations

Single Elimination (Common)

One pair digit is blocked in key chute cells, so remove the other digit from valid targets.

Double Elimination (Rare)

Both pair digits are blocked in key chute cells, so valid targets can lose both digits.

Advanced positions can produce extra either-end eliminations. Verify them carefully before removing candidates.

Modern solvers often classify this as a special case of AIC/chaining.

Related Techniques

Related Techniques

🔗XY-Chain🧠AIC🎨Simple Coloring
Alternating Inference ChainsExpert

Alternating Inference Chains

See the broader chain framework behind contradiction-based eliminations.