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A calm bright study with an abacus, wooden cones and cubes and a small grid of square tiles with tiny flags, a teenager reasoning carefully about which tile is safe

Proof Sweeper

Every square you open has to come with a reason. You can’t lose — you can only prove.

📜 Prove a board →

Three rules of inference

  • done. A number already touches all its mines → its other hidden neighbours are safe.
  • full. A number’s undecided neighbours are exactly the mines it still needs → they are all mines.
  • subset. Number A’s undecided squares all sit inside number B’s → B’s extra squares hold exactly (B’s missing mines − A’s): if that’s 0 they are safe; if it equals their count, they are mines.

Only mines you have proven count as premises later — a hunch is not a fact.

Circuit levels

In 2000 Richard Kaye showed Minesweeper consistency is NP-complete by building boards that behave like logic circuits — “wires” that carry a true/false value and “gates” that combine them. Proof Sweeper has four of them: a wire, a NOT gate, an AND gate and an OR gate. You are given the inputs, and you prove what the output must be.

Two more levels run a gate backwards: you are given the output and prove what the inputs must be. Only two outputs make that possible — an AND gate whose output is a mine, and an OR gate whose output is safe. Every other output leaves several possible inputs. Working backwards is the hard direction: asking whether any inputs could produce an output is the satisfiability problem, and Kaye's gadgets turn every such question into a Minesweeper board.

Try a circuit level →

A case · you decide

A case: Lucia and the word “obviously”

Lucia — a 17-year-old whose geometry teacher keeps writing “why?” in the margin

The idea in play: a claim is only proven when you can name the rule and the facts that force it.

Lucia is quick at Minesweeper. Her teacher bets her she can’t clear a board if every move has to be justified in writing.

The first moves are easy: a 1 with only one hidden neighbour means that neighbour is a mine. She writes it down — rule full, premise B2=1.

Then she reaches a spot where she just “knows” D4 is safe. The checker asks for the premise. She has to stop and find it.

D4 feels safe. What does she write?

No single number decides the next square. What now?

Write a proof →

Where this game comes from

Mine-clearing number puzzles

Grid games where numbers tell you how many hidden mines are next door appeared on home computers in the early 1980s. The best-known version was made by Curt Johnson, and ported to Windows by Robert Donner; it came out in a Microsoft Entertainment Pack in 1990 and then with Windows 3.1 in 1992. In that classic game some boards force a guess, often a coin toss between two squares.

One thing to hold: Each number is a promise about the eight squares around it, and nothing else.

Why it matters: In 2000 the mathematician Richard Kaye proved that deciding whether a board’s numbers can all be true is among the hardest problems in a famous class called NP (it is “NP-complete”). A quick way to check any board would answer one of the biggest open questions in mathematics.

What we don’t know: Who wrote the very first one is still argued about; several early games had the same idea.

Dates:

  1. 1990 Curt Johnson and Robert Donner’s game ships in a Microsoft Entertainment Pack.
  2. 1992 It comes with Windows 3.1.
  3. 2000 Richard Kaye proves the board-checking problem is NP-complete.

Sources: Richard Kaye, “Minesweeper is NP-complete”, The Mathematical Intelligencer 22 (2000) · Microsoft Windows 3.1 (1992) · Wikipedia, “Microsoft Minesweeper” (read 2026-10-20)

What next? Pick a door

Chosen from what this app is about. What you open is remembered on this device only.