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# domino

A domino is a small rectangular tile, marked with spots, that is used to play games. There are many varieties of dominoes and games based on them. Most domino tiles, however, have roughly 2-to-1 proportions and each half of each domino has spots arranged as on a six-sided die; the set generally contains tiles with all possible combinations of two numbers. A tile is identified by the number of spots on each half: for example, "1-6" or "3-3." English/American dominoes include blank sides; Chinese dominoes don't but do duplicate some whole tiles. English/American dominoes can also be bought in larger sets with numbers of spots up to nine or twelve per side.

Other than games of strategy, there are many mathematical puzzles that involve dominoes. Some of these puzzles involve tiling variations on the standard 8 × 8 chessboard. Of course, a standard board can easily be tiled – by using four dominos in each row. But what if two squares are removed, one each from diagonally opposite corners of the chessboard? Can this reduced board be completely tiled by non-overlapping dominos? No! Suppose it were possible to totally cover the modified chessboard with non-overlapping dominos. In any complete tiling, every domino must cover exactly one white square and one black square. Thus the modified board must have exactly the same number of black and white squares. But the two removed squares, from diagonally opposite corners of a chessboard, must be same color. Since there can't be the same number of white squares and black squares on the modified board it must be impossible to tile the modified board with non-overlapping dominos.

Another common pastime using domino tiles is to stand them on edge in long lines, then topple the first tile, which falls on and topples the second, etc., resulting in all of the tiles falling. Arrangements of thousands of tiles have been made that take several minutes to fall. By analogy, phenomena of chains of small events each causing similar events leading to eventual catastrophe are called domino effects.

The word "domino" was first used to refer to the hooded black cape worn by priests, and later to black masks (of the Lone Ranger type) worn at masquerade balls. The domino is the simplest form of polyomino.

## Domino problem

The domino problem asks if there is an algorithm (a set of instructions) that, when given a particular shape as an input, decides if the shape can be used to tile the entire plane. The solution to this unresolved problem is tied up with Heesch numbers. The domino problem in turn has a deep connection with the Einstein problem.