Daily Challenges

The 1-3-1 Corner Pattern: Mines at the Flanks

The 1-3-1 corner pattern appears when a “3” sits at a wall corner with a “1” on each side. The corner geometry (only 3–5 neighbors) makes this a tight constraint — the cells flanking the “3” are mines, and the remaining cell adjacent to the “3” is safe.

Also known as: 1-3-1 corner. This is the corner variant of the classic 1-2-1 pattern — the “3” plays the role of a “2” after reduction accounts for the extra neighbor created by the corner bend.

How the 1-3-1 Corner Works

The Setup

Imagine a wall that turns a corner. The corner cell itself is revealed, and three numbers run along the wall: 1, 3, 1 — with the “3” at or near the corner.

    W  W  ?  ?
    W  1  ?  ?
    ?  3  ?  ?
    ?  1  W  W
    ?  ?  W  W

The Logic

  1. The corner “3” touches several unrevealed cells, but only a subset are shared with each adjacent “1”.
  2. Each “1” needs exactly one mine among its unrevealed neighbors.
  3. The “3” needs three mines among its neighbors. The two cells it shares with the “1"s must account for one mine each (from the “1” constraints). That’s two mines placed. The third mine must be elsewhere in the “3”’s neighborhood.
  4. The cells between each “1” and the “3” (the flanking cells) are where the mines go. The remaining cell adjacent only to the “3” is safe.

Why This Is a Corner Pattern

On a straight wall, 1-3-1 doesn’t resolve the same way — the “3” has too many neighbors. At a corner, the cell count drops, creating the tight constraint that forces a unique solution. The corner geometry is what makes this pattern work.

Variations

1-3-1 With Flags Nearby

If the “3” already has one flag adjacent, it reduces to an effective “2” — and 1-2-1 logic applies to the remaining cells. This is reduction revealing a simpler pattern.

1-3-1 on a Concave Corner

When the wall bends inward (concave), the “3” has even fewer unrevealed neighbors, making the pattern even more constrained. In some configurations, the entire neighborhood is solvable from the “3” alone.

Extended: 2-3-1 or 1-3-2 at Corners

If one of the “1"s is a “2” instead, the flanking mine on that side is still guaranteed, but the other side requires more analysis. Apply subset logic to determine the remaining cells.

How to Spot It

  1. Look at corner cells with a “3.” Three is a high number — when it appears near a corner, it’s almost always solvable.
  2. Check the flanking numbers. If both sides show “1”, you have the 1-3-1 corner.
  3. Count unrevealed neighbors. Confirm the “3” has few enough unrevealed neighbors for the constraint to be tight.

Common Mistakes

Confusing With Straight-Wall 1-3-1

On a straight wall with no corner, 1-3-1 does NOT resolve the same way — the “3” has more neighbors and the pattern is ambiguous. The corner is essential.

Forgetting to Reduce First

If the “3” already has a flagged neighbor, reduce it to “2” first, then apply 1-2-1 pattern logic. Applying 1-3-1 corner logic to an already-partially-solved “3” leads to incorrect deductions.