Daily Challenges

Subset Logic: Finding Mines

Subset logic for safe cells works when two overlapping numbers need the same mine count. But when the larger set needs more mines than the subset accounts for, the extra cells must contain the difference. Those extra cells are confirmed mines.

Also known as: H3 (hidden mine pattern). In shorthand notation, “H” patterns refer to hidden deductions from overlapping constraint regions. H3 specifically finds mines rather than safe cells.

The General Rule

Given two numbered cells:

  • Cell A needs $m$ mines among unrevealed neighbors ${S_A}$
  • Cell B needs $n$ mines among unrevealed neighbors ${S_B}$
  • ${S_A} \subseteq {S_B}$ (A’s neighbors are a subset of B’s)
  • $n > m$ (B needs more mines than A)

Then the non-overlapping cells ${S_B \setminus S_A}$ contain exactly $n - m$ mines.

If $n - m$ equals the number of non-overlapping cells, every extra cell is a mine. Flag them all.

Classic Example: A “1” and a “3”

A “1” touches two unrevealed cells: {A, B}. A neighboring “3” touches four unrevealed cells: {A, B, C, D}.

  • “1” needs 1 mine in {A, B}.
  • “3” needs 3 mines in {A, B, C, D}.
  • {A, B} ⊂ {A, B, C, D}, and $3 - 1 = 2$ mines in {C, D}.
  • {C, D} has exactly 2 cells and 2 mines → both C and D are mines.

Example: A “2” and a “3”

A “2” touches three unrevealed cells: {P, Q, R}. A “3” touches four unrevealed cells: {P, Q, R, S}.

  • “2” needs 2 mines in {P, Q, R}.
  • “3” needs 3 mines in {P, Q, R, S}.
  • $3 - 2 = 1$ mine in {S}.
  • {S} has 1 cell and 1 mine → S is a mine.

When the Difference Doesn’t Fill the Extra Cells

Sometimes $n - m$ is positive but less than the number of extra cells. For example:

  • “1” needs 1 mine in {A, B}.
  • “2” needs 2 mines in {A, B, C, D, E}.
  • $2 - 1 = 1$ mine in {C, D, E}.

You know one of {C, D, E} is a mine but not which one. This does not give you a definitive action on any single cell. However, it does create a new constraint (1 mine in 3 cells) that may combine with other numbers to produce a definitive answer.

Recognizing the Mine-Finding Subset

Step 1: Find a Tight Number with Low Mine Count

Corner “1"s, edge “1"s, and numbers almost satisfied by flags create tight constraints with low remaining mine counts.

Step 2: Check Adjacent Numbers with Higher Counts

If a neighboring number needs more mines and its unrevealed set contains the tight number’s set, you have a mine-finding opportunity.

Step 3: Compute the Difference

Subtract: (larger count) − (smaller count) = mines in the extra cells. If this equals the number of extra cells, flag them all.

Relationship to 1-2-X

The 1-2-X pattern is a specific instance of mine-finding subsets:

  • “1” needs 1 mine in {A, B}
  • “2” needs 2 mines in {A, B, X}
  • $2 - 1 = 1$ mine in {X} → X is a mine

Every 1-2-X identification is a subset calculation. Learning the general principle lets you apply the same logic to any number combination, not just 1 and 2.

Practice Tips

  1. After flagging with 1-2-X, look for more. The flag you just placed may reduce an adjacent number, creating a new subset opportunity.
  2. Compare every pair of adjacent numbers. Ask: “Do these share unrevealed cells? Is one a subset of the other? What’s the difference?” This systematic approach catches patterns that pure visual recognition might miss.
  3. Don’t forget flags. Always subtract existing flags before comparing. A “4” with two adjacent flags is effectively a “2” for this purpose.