If you play Minesweeper on Cool Math Games, this independent guide offers practice with minesweeper probability. Its exercises distinguish a proved local deduction from an estimate that depends on additional assumptions.
When logic does not identify a safe square, a probability statement needs assumptions. A fraction based on one clue is not automatically the exact risk of a square on the complete board.
A practical starting point
The examples below count assignments within a simplified local set. They assume the listed arrangements are equally likely. Other clues and a known total-mine count can change the possibilities on a real board.
Count arrangements under declared assumptions
A local probability can be calculated by counting possible mine assignments when their relative likelihoods are specified. The examples assume all assignments with the stated mine count inside the listed set are equally likely. They do not include undisplayed clues or a complete-board mine total. On a real board, either could change the allowed arrangements. A nonzero risk remains a risk even when it is smaller than another estimate. Keep the model’s assumptions beside the fraction instead of presenting it as guaranteed safety.
A mistake to watch for
Never turn a lower estimated risk into a guarantee. A decision can be reasonable under a model and still hit a mine. Keep the model, its limitations and the outcome separate.
Worked examples and practice decisions
Try each question before reading its explanation. These examples are deliberately limited to the information they state. They separate a rule or calculation from a broader judgement about a complete game.
1. Clue exercise 1: check the unknown set
A simplified local set contains 6 covered cells and exactly 3 mines. Assume every placement of those mines within this set is equally likely and no other clue changes the possibilities. Before reading the calculation, separate the covered-cell set from any nearby square that is not part of it. All confirmed mines in this exercise are assumptions stated in the question, not guesses represented by flags.
There are 20 allowed placements. A chosen cell contains a mine in 10 of them, giving a local risk of 10/20, equal to 3/6. This is a conditional model. It is not an exact full-board probability if other information changes the allowed placements.
Countercheck: there are 20 locally allowed assignments in this model. Add a hypothetical extra clue that rules out some assignments and the equal-risk conclusion may change. That is why a full-board probability needs more information than a single local count. State what information you have included before comparing fractions from different regions.
2. Clue exercise 2: check the unknown set
A simplified local set contains 4 covered cells and exactly 3 mines. Assume every placement of those mines within this set is equally likely and no other clue changes the possibilities. Before reading the calculation, separate the covered-cell set from any nearby square that is not part of it. All confirmed mines in this exercise are assumptions stated in the question, not guesses represented by flags.
There are 4 allowed placements. A chosen cell contains a mine in 3 of them, giving a local risk of 3/4, equal to 3/4. This is a conditional model. It is not an exact full-board probability if other information changes the allowed placements.
Countercheck: there are 4 locally allowed assignments in this model. Add a hypothetical extra clue that rules out some assignments and the equal-risk conclusion may change. That is why a full-board probability needs more information than a single local count. State what information you have included before comparing fractions from different regions.
3. Clue exercise 3: check the unknown set
A simplified local set contains 7 covered cells and exactly 1 mines. Assume every placement of those mines within this set is equally likely and no other clue changes the possibilities. Before reading the calculation, separate the covered-cell set from any nearby square that is not part of it. All confirmed mines in this exercise are assumptions stated in the question, not guesses represented by flags.
There are 7 allowed placements. A chosen cell contains a mine in 1 of them, giving a local risk of 1/7, equal to 1/7. This is a conditional model. It is not an exact full-board probability if other information changes the allowed placements.
Countercheck: there are 7 locally allowed assignments in this model. Add a hypothetical extra clue that rules out some assignments and the equal-risk conclusion may change. That is why a full-board probability needs more information than a single local count. State what information you have included before comparing fractions from different regions.
4. Clue exercise 4: check the unknown set
A simplified local set contains 6 covered cells and exactly 5 mines. Assume every placement of those mines within this set is equally likely and no other clue changes the possibilities. Before reading the calculation, separate the covered-cell set from any nearby square that is not part of it. All confirmed mines in this exercise are assumptions stated in the question, not guesses represented by flags.
There are 6 allowed placements. A chosen cell contains a mine in 5 of them, giving a local risk of 5/6, equal to 5/6. This is a conditional model. It is not an exact full-board probability if other information changes the allowed placements.
Countercheck: there are 6 locally allowed assignments in this model. Add a hypothetical extra clue that rules out some assignments and the equal-risk conclusion may change. That is why a full-board probability needs more information than a single local count. State what information you have included before comparing fractions from different regions.
5. Clue exercise 5: check the unknown set
A simplified local set contains 5 covered cells and exactly 2 mines. Assume every placement of those mines within this set is equally likely and no other clue changes the possibilities. Before reading the calculation, separate the covered-cell set from any nearby square that is not part of it. All confirmed mines in this exercise are assumptions stated in the question, not guesses represented by flags.
There are 10 allowed placements. A chosen cell contains a mine in 4 of them, giving a local risk of 4/10, equal to 2/5. This is a conditional model. It is not an exact full-board probability if other information changes the allowed placements.
Countercheck: there are 10 locally allowed assignments in this model. Add a hypothetical extra clue that rules out some assignments and the equal-risk conclusion may change. That is why a full-board probability needs more information than a single local count. State what information you have included before comparing fractions from different regions.
6. Clue exercise 6: check the unknown set
A simplified local set contains 3 covered cells and exactly 1 mines. Assume every placement of those mines within this set is equally likely and no other clue changes the possibilities. Before reading the calculation, separate the covered-cell set from any nearby square that is not part of it. All confirmed mines in this exercise are assumptions stated in the question, not guesses represented by flags.
There are 3 allowed placements. A chosen cell contains a mine in 1 of them, giving a local risk of 1/3, equal to 1/3. This is a conditional model. It is not an exact full-board probability if other information changes the allowed placements.
Countercheck: there are 3 locally allowed assignments in this model. Add a hypothetical extra clue that rules out some assignments and the equal-risk conclusion may change. That is why a full-board probability needs more information than a single local count. State what information you have included before comparing fractions from different regions.
7. Clue exercise 7: check the unknown set
A simplified local set contains 8 covered cells and exactly 3 mines. Assume every placement of those mines within this set is equally likely and no other clue changes the possibilities. Before reading the calculation, separate the covered-cell set from any nearby square that is not part of it. All confirmed mines in this exercise are assumptions stated in the question, not guesses represented by flags.
There are 56 allowed placements. A chosen cell contains a mine in 21 of them, giving a local risk of 21/56, equal to 3/8. This is a conditional model. It is not an exact full-board probability if other information changes the allowed placements.
Countercheck: there are 56 locally allowed assignments in this model. Add a hypothetical extra clue that rules out some assignments and the equal-risk conclusion may change. That is why a full-board probability needs more information than a single local count. State what information you have included before comparing fractions from different regions.
8. Clue exercise 8: check the unknown set
A simplified local set contains 3 covered cells and exactly 2 mines. Assume every placement of those mines within this set is equally likely and no other clue changes the possibilities. Before reading the calculation, separate the covered-cell set from any nearby square that is not part of it. All confirmed mines in this exercise are assumptions stated in the question, not guesses represented by flags.
There are 3 allowed placements. A chosen cell contains a mine in 2 of them, giving a local risk of 2/3, equal to 2/3. This is a conditional model. It is not an exact full-board probability if other information changes the allowed placements.
Countercheck: there are 3 locally allowed assignments in this model. Add a hypothetical extra clue that rules out some assignments and the equal-risk conclusion may change. That is why a full-board probability needs more information than a single local count. State what information you have included before comparing fractions from different regions.
9. Clue exercise 9: check the unknown set
A simplified local set contains 7 covered cells and exactly 3 mines. Assume every placement of those mines within this set is equally likely and no other clue changes the possibilities. Before reading the calculation, separate the covered-cell set from any nearby square that is not part of it. All confirmed mines in this exercise are assumptions stated in the question, not guesses represented by flags.
There are 35 allowed placements. A chosen cell contains a mine in 15 of them, giving a local risk of 15/35, equal to 3/7. This is a conditional model. It is not an exact full-board probability if other information changes the allowed placements.
Countercheck: there are 35 locally allowed assignments in this model. Add a hypothetical extra clue that rules out some assignments and the equal-risk conclusion may change. That is why a full-board probability needs more information than a single local count. State what information you have included before comparing fractions from different regions.
10. Clue exercise 10: check the unknown set
A simplified local set contains 5 covered cells and exactly 3 mines. Assume every placement of those mines within this set is equally likely and no other clue changes the possibilities. Before reading the calculation, separate the covered-cell set from any nearby square that is not part of it. All confirmed mines in this exercise are assumptions stated in the question, not guesses represented by flags.
There are 10 allowed placements. A chosen cell contains a mine in 6 of them, giving a local risk of 6/10, equal to 3/5. This is a conditional model. It is not an exact full-board probability if other information changes the allowed placements.
Countercheck: there are 10 locally allowed assignments in this model. Add a hypothetical extra clue that rules out some assignments and the equal-risk conclusion may change. That is why a full-board probability needs more information than a single local count. State what information you have included before comparing fractions from different regions.
Use the guide in your next session
Before comparing two fractions, ask whether they were calculated under compatible assumptions. A tidy number is not useful if one estimate ignores information that the other includes.
Begin with one observation connected to minesweeper probability. After checking an example, close the answer and reconstruct the reasoning in your own words. If a step is uncertain, return to that step rather than repeating a result you cannot explain. A small, specific correction provides a clearer next aim than trying to improve every part of a game at once.
Frequently asked questions
Does a flag prove a mine is present?
A flag records a choice. Subtract it from a clue only when the reasoning has actually established that mine.
Are diagonal cells neighbours?
Yes. Interior clues can refer to eight adjacent cells, while edges and corners have fewer neighbours.
Does a local probability give exact full-board risk?
Only if its assumptions match all relevant information. Other clues or a known total mine count can alter the possible assignments.
More cool math games guides
- Minesweeper Beginner Guide: Read the Eight Neighbours
- Minesweeper Safe Squares: Subtract Confirmed Flags
- Open the relevant game or guide page
Sources and scope
This is an independent, AI-assisted editorial guide. The worked examples and diagrams are original teaching material, not official game screenshots or a claim of personal live-game testing. Game interfaces can change; check the current instructions when controls or rules differ.
