Almost Locked Sets (ALS) in Sudoku: How to Use the ALS-XZ Rule to Eliminate Candidates

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Published April 16, 2026 · 12 min read

Almost Locked Sets (ALS) represent one of the most powerful expert-level techniques in Sudoku. While techniques like X-Wing or Swordfish focus on a single digit, ALS works across multiple digits simultaneously, making it exceptionally versatile.

The core idea: find two groups of cells (each an “Almost Locked Set”) that are linked by a shared candidate called the Restricted Common Candidate (RCC). A second shared candidate can then be eliminated from cells that see both ALS.

In this guide you will learn what an ALS is, understand the ALS-XZ elimination rule, walk through a detailed example, and discover how to spot these patterns in your own puzzles.

✅ What Is an Almost Locked Set?

An Almost Locked Set (ALS) is a group of N cells within a single house (row, column, or box) that together contain exactly N+1 different candidates.

Examples by size:

• 1 cell, 2 candidates: A bivalue cell like R3C5={4,7} is the simplest ALS.
• 2 cells, 3 candidates: For instance R1C1={8,9} and R1C2={1,8,9} in the same row — together they hold {1,8,9}.
• 3 cells, 4 candidates: Three cells sharing four digits — one step above a Naked Triple.

The word “almost” is key: a Locked Set has N cells with N candidates and is fully determined. An ALS has one extra candidate, so it is “almost” locked. This surplus candidate is what makes ALS useful for elimination logic.

🔗 The ALS-XZ Rule

The ALS-XZ rule is the fundamental elimination technique using Almost Locked Sets. It requires two ALS linked by a special candidate.

Why it works: Because x is restricted, at most one of the two ALS can keep it. Whichever ALS loses x becomes fully locked (N cells, N candidates). In either scenario, the surviving candidates of that ALS are determined. Candidate z must be in one ALS or the other — it cannot be in any cell that sees all z-cells in both.

In plain language:

1. Find two ALS (A and B) with no overlapping cells.
2. Identify a shared candidate x whose cells in ALS A all see the x-cells in ALS B (this is the RCC).
3. Find another shared candidate z.
4. Eliminate z from any cell (outside both ALS) that can see every z-cell in ALS A and every z-cell in ALS B.

🔒 The Restricted Common Candidate (RCC)

The Restricted Common Candidate (RCC) is the linchpin of the ALS-XZ rule. It is the candidate digit labelled x that links ALS A and ALS B together.

Typical RCC connections:

• Same column: An x-cell in ALS A and an x-cell in ALS B share a column.
• Same box: All x-cells in both ALS belong to (or see into) the same box.
• Same row: x-cells line up in the same row across the two ALS.

If x had cells in ALS A and ALS B that did not see each other, both ALS could potentially keep x, and the elimination logic would break down.

🔎 Step-by-Step Example

Let’s work through a real ALS-XZ elimination. We will identify two ALS, find the RCC, and eliminate a candidate.

Step 1: Identify ALS A

Look at Row 1. Two unsolved cells in Box 1 form an ALS:

• R1C1 = {8,9}
• R1C2 = {1,8,9}

Together they hold 3 candidates {1,8,9} across 2 cells — that is N=2 cells with N+1=3 candidates. This is ALS A.

Step 2: Identify ALS B

Now look at Row 2. Three unsolved cells form another ALS:

• R2C3 = {3,8}
• R2C4 = {3,9}
• R2C5 = {3,7,9}

Together they hold 4 candidates {3,7,8,9} across 3 cells — N=3, N+1=4. This is ALS B.

Step 3: Find the Restricted Common Candidate (RCC)

ALS A has candidates {1,8,9}. ALS B has {3,7,8,9}. The common candidates are 8 and 9.

Check candidate 8 as the RCC:

• 8 in ALS A: R1C1 and R1C2 (both in Row 1, Box 1).
• 8 in ALS B: only R2C3 (Row 2, Box 1).
• R1C1 sees R2C3 through Box 1 ✔. R1C2 sees R2C3 through Box 1 ✔.

All x-cells in ALS A see all x-cells in ALS B. Candidate 8 is a valid RCC (x=8).

Step 4: Identify the Elimination Candidate (z)

The other common candidate is 9 — this is our z.

• 9 in ALS A: R1C1 and R1C2 (Row 1, Box 1).
• 9 in ALS B: R2C4 and R2C5 (Row 2, Box 2).

Step 5: Eliminate z

Remove candidate 9 from every cell outside both ALS that sees all four z-cells (R1C1, R1C2, R2C4, R2C5):

• R1C4 — sees R1C1 & R1C2 (Row 1) and R2C4 & R2C5 (Box 2) — {3,5,6,9} → {3,5,6}
• R1C5 — sees R1C1 & R1C2 (Row 1) and R2C4 (Col 4... wait — actually Box 2) & R2C5 (Col 5) — {3,6,9} → {3,6}
• R1C6 — sees R1C1 & R1C2 (Row 1) and R2C4 & R2C5 (Box 2) — {3,5,6,9} → {3,5,6}

Three cells lose candidate 9!

Step 6: Result

After removing candidate 9 from three cells in Row 1, their candidate lists shrink significantly. R1C5 now has only {3,6}, which may enable Naked Pairs or Hidden Singles to make further progress.

⚖️ ALS vs. Simpler Techniques

ALS-XZ is more powerful than most single-digit techniques because it works with multiple digits at once. An XY-Wing is actually a special case of ALS-XZ where both ALS are bivalue cells.

🕵️ How to Spot ALS-XZ Patterns

Practical tips:

• Start small: Bivalue cells are the easiest ALS to spot. Pair a bivalue cell with a 2-cell ALS in an overlapping box for a quick ALS-XZ.
• Focus on box boundaries: The RCC often links through a shared box, so look at cells near box borders.
• Use pencil marks: ALS detection requires complete and accurate candidate lists. Make sure your pencil marks are fully up to date.
• Look at congested areas: Regions with many unsolved cells and overlapping candidates are fertile ground for ALS patterns.

⚠️ Common Mistakes

1. Forgetting the House Constraint

Each ALS must have all its cells in a single house (one row, one column, or one box). An arbitrary collection of cells is not an ALS unless they share a house.

2. Missing the RCC Visibility

The Restricted Common Candidate requires that every cell with x in ALS A sees every cell with x in ALS B. If even one pair doesn’t share a row, column, or box, the RCC is invalid.

3. Eliminating the Wrong Candidate

Only the non-RCC common candidate (z) can be eliminated. The RCC (x) itself is not eliminated — it is the link that makes the logic work.

4. Including Cells from Both ALS in Eliminations

Eliminations only apply to cells outside both ALS. Do not remove z from cells that belong to ALS A or ALS B — those cells are part of the pattern.

5. Incomplete Pencil Marks

ALS detection depends entirely on accurate candidate lists. Missing a candidate or having an extra one leads to incorrect ALS identification and wrong eliminations.

📅 When to Look for ALS

1. Basic: Naked Singles, Hidden Singles, Full House.
2. Early intermediate: Locked Candidates, Pointing Pairs, Box/Line Reduction.
3. Intermediate: Naked Pairs, Hidden Pairs, Naked Triples.
4. Advanced: X-Wing, XY-Wing, W-Wing, Skyscraper, Simple Colouring.
5. Expert: ALS-XZ, Swordfish, Jellyfish, Unique Rectangles.

Use ALS-XZ when simpler techniques are exhausted. It is one of the last techniques to try before resorting to advanced chains or trial-and-error.

🚀 Beyond ALS-XZ

ALS-XZ is the gateway to ALS-based techniques. Once you are comfortable spotting two linked ALS, extending to three (ALS-XY-Wing) or longer chains is a natural next step. The underlying logic remains the same: restricted common candidates force one side or the other to lock, and the resulting certainty drives eliminations.

🎯 Practice ALS-XZ