Penrose minesweeper
Ordinary minesweeper rules on tilings that are not square grids: the eleven Archimedean tilings, three Penrose tilings, a pair of quasicrystals with eight- and twelve-fold symmetry, a substitution tiling, and the cells of a random point set. Two tiles are neighbours when they share at least one corner. On the aperiodic boards the pattern never repeats, so no two neighbourhoods look quite the same.
15.0% mine density
Controls
- Left click — reveal a tile.
- Right click — place or remove a flag.
- Left + right together (or middle click, or double click) on a revealed number — chord: if you have flagged exactly that many neighbours, every other neighbour is revealed at once.
- Hover a tile to outline the tiles that count as its neighbours — worth leaning on, since corner-sharing is much harder to eyeball than a square grid.
- Scroll to zoom, drag to pan.
- Touch — tap to reveal, long press to flag, tap a number to chord, pinch to zoom. Or switch on flag mode and tap to flag.
Rules
Every tile either hides a mine or, when revealed, shows how many of its neighbours are mined. A tile with no mined neighbours opens up and keeps opening its neighbours until it reaches tiles that do touch a mine. Clear every tile that is not a mine and you win. Your first click is always safe.
With no guessing switched on, the board is dealt only after your first click and is checked by a solver first: every tile can be worked out by logic, so you never have to flip a coin — including at the very end, where the mine counter is part of the reasoning. Switch it off for a plain random board. On most lattices a guess-free layout is found in well under a second; on the densest and most highly connected boards the search sometimes runs out of time, and says so rather than pretending.
Lattices
Eleven of the eighteen are the Archimedean tilings — every edge-to-edge tiling by regular polygons in which all vertices look alike — named by the polygons around a vertex, so 3.4.6.4 means triangle, square, hexagon, square.
The rest do not repeat. Penrose P3 uses fat and thin rhombi and comes from de Bruijn's pentagrid; on a large board the ratio of the two approaches the golden ratio, φ ≈ 1.618. Penrose P2 is the original kites and darts, grown by repeated subdivision, and its kite-to-dart ratio approaches φ as well. Widening the pentagrid to four and six line families gives the Ammann–Beenker tiling, an eight-fold quasicrystal of squares and 45° rhombs whose rhomb-to-square ratio is √2, and a dodecagonal twelve-fold one. The pinwheel is Conway and Radin's tiling by a single 1-2-√5 right triangle, which appears in unboundedly many orientations. Voronoi is the cells of a random point set, so it has no repeating structure at all and is different every game.
The grid offsets in those constructions decide how symmetric the result is. Equal offsets make the pentagrid commute with a 72° rotation, so a P3 board comes out symmetric — but which symmetry depends on the value, which is why there are two P3 entries. Offsets of ⅕ sum to a whole number, which is de Bruijn's condition for a genuine Penrose tiling obeying the matching rules, and give a five-fold rosette; offsets of ½ give a ten-fold one. For an even number of line families one family lies at 90° and reflects onto itself, which buys a single mirror rather than full rotational symmetry. Every board is turned so that a mirror line stands vertical.
Tiles per board are the same on every lattice, but the number of neighbours is not: it runs from 6 on the hexagonal tiling to 13 on the pinwheel. Mine counts are set so that the average number of mines per clue matches across lattices, which is what actually governs how hard a board is to read. On the square lattice that reproduces classic minesweeper's own densities almost exactly.