When Minesweeper makes you guess

First, be certain you have to guess

Most guesses are unnecessary. Before accepting one, walk the entire boundary again and check three things at every number: whether its flags already satisfy it, whether its unopened neighbours exactly equal what it still needs, and whether subtracting its flags turns it into a pattern you recognise.

The patterns guide covers the shapes worth knowing. A great many “forced” guesses are a 1-1 pattern nobody looked for.

Working out the local odds

When a guess really is forced, it is rarely a coin flip everywhere. Different cells carry different risk, and the numbers touching them tell you how much.

The rough method, which is enough for almost every position:

1. For each unopened cell, find the numbers touching it. Ignore cells touching nothing — they are governed only by the global count.
2. For each of those numbers, work out how many mines it still needs and how many unopened cells it still touches. The ratio is that number's estimate of the risk.
3. Take the worst estimate for the cell. A cell that looks like one-in-three to one number and one-in-two to another is one-in-two: the tighter constraint wins.
4. Open the cell with the lowest worst-case risk.

The mine counter is a clue too

People treat the remaining-mine number as a progress bar. It is actually a constraint over the entire board, and late in a game it is often the strongest one available.

Suppose eight mines remain and there are forty unopened cells. Cells far from any number carry a flat one-in-five risk. If the boundary is tense — a cluster of 3s with few unopened neighbours — then a cell out in the open field can easily be several times safer than anything on the edge.

The reverse happens too, and catches people out. When two mines remain among four unopened cells, the open field is now one-in-two and the boundary, with its numbers to constrain it, may well contain a cell you can prove safe. Late in a board, the field gets more dangerous while the boundary gets more informative.

Sometimes the global count resolves a position outright. If the numbers along the boundary can only be satisfied by using every remaining mine, then every cell away from the boundary is safe, and you can open all of them.

Choose a guess that teaches you something

Two cells with the same risk are not equally good. What matters afterwards is how much you learn, and that varies enormously.

• Guess into the largest unopened region. A safe result there cascades open and hands you a dozen new numbers. A safe result on a tight boundary often gives you one number and the same dilemma.
• Prefer corners and edges when risk is equal. Fewer neighbours means the number you reveal is more constrained, and more likely to be immediately useful.
• Avoid guessing into a spot that only resolves itself. If a cell's outcome tells you nothing about anything else, you have spent your risk badly.
• Guess early rather than late. Risk is roughly constant, but the value of what you learn falls as the board fills. Postponing a necessary guess usually makes it worse.

What a realistic success rate looks like

A large board will typically force one or two genuine guesses. At one-in-three each, that alone caps how often a careful player finishes, and no technique changes it. This is worth knowing because it stops you treating a loss as a mistake.

What it does mean is that the losses you should care about are the avoidable ones. When a run ends, the useful question is whether you hit a real coin flip or whether you mis-flagged something twenty moves earlier and everything after it was built on a wrong assumption. The second kind is much more common than people expect.

Chording — clicking a satisfied number to open all its neighbours — is the usual way a bad flag turns fatal, because it commits to your flags all at once. If a run keeps ending during a chord, the problem is the flagging, not the luck.

Our first click is always safe, so nothing is lost before the board has told you anything. Everything after that is yours.