| C1 | C2 | C3 | C4 | |
|---|---|---|---|---|
| R1 | ||||
| R2 | ||||
| R3 | ||||
| R4 |
With one mine on the whole board, is R4C4 safe?
Make a prediction before opening the reasoning.
Compare the predictions
- Yes: R4C4 is proved safe.
Supported. The only mine must be beside R1C1, leaving no mine for the distant corner.
- No conclusion is possible.
Not supported. That would be true without the global total. The stated total settles all distant squares.
- R4C4 must be the mine.
Not supported. That would leave no mine beside the revealed 1.
Predictions stay on this page; they are not saved or sent.
The position and the question
The total of one mine is part of the problem statement, not a count inferred from player flags. R1C1 is the only revealed square. Its three neighbours are R1C2, R2C1 and R2C2, so those three squares contain exactly one mine between them.
Do not inspect the stored mine location when reasoning. We can decide the distant corner using only that revealed 1 and the whole-board total. The clue’s neighbouring squares remain genuinely unresolved.
Show the reasoning
Reserve the entire mine budget
The three neighbours of R1C1 need one mine. The whole board contains one mine. Subtracting the required local mine from the total leaves zero mines for every square outside that three-square group and the revealed clue.
Name what is proved and what is not
All 12 other hidden squares are safe, including R4C4. However, any one of R1C2, R2C1 or R2C2 could hold the mine. There are three complete mine placements compatible with the current information. A safe distant square and an unresolved nearby group can coexist.
Choose a justified action
Reveal a proved-safe square such as R4C4 rather than placing an unsupported flag beside the 1. New revealed numbers may later distinguish the nearby candidates. A flag records a player’s belief; it does not supply a new mine constraint.
Numbers count all eight neighbours C1 C2 C3 C4 R1 R2 R3 R4 After the proved-safe R4C4 is revealed. Its new 0 was not needed for the earlier safety proof. Change one fact and repeat the argument
If the board total were two mines, the neighbouring group would still contain exactly one. The second mine would lie in one of the 12 distant squares. There would be 3 × 12 = 36 compatible placements, and R4C4 would no longer be proved safe. The shape of the visible clue has not changed; the global budget has.
The tempting shortcut
Only squares beside a revealed clue can be proved safe.
Where the shortcut breaks
The total mine count applies to the entire board. Once the frontier requires the whole budget, every outside square is safe.
A better decision
Combine local clue requirements with the declared total, keeping flags separate from proved mines.
Changed conditions
Keep the visible 1 but remove the whole-board mine total from the information supplied. Is the distant corner still guaranteed safe?
Check the changed-condition answer
No. A placement with the nearby required mine and an additional mine at R4C4 satisfies the only visible clue. A placement without that distant mine also satisfies it. The total was essential to the safety proof.
Take this back to your next game
State the full information used in a deduction. Changing one global constraint can reverse a local-looking conclusion.
How the claim was checked
Enumerate all hidden-cell assignments consistent with the revealed clue and stated mine total.
- Total 1: exactly 3 compatible placements
- Every distant square safe across all 3
- Total 2: 36 compatible placements; each distant square mined in 3
Read the fixed positions and supporting proof data. Computational checks establish only the claim and position stated here. They do not establish a player difficulty rating or make an unproved strategy optimal.
Playfield Arcade · Updated 4 October 2026 · Original case design, AI-assisted writing and code, with reproducible position checks. Our method · Report a correction