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Chessboard paradox. The chessboard paradox [1] [2] or paradox of Loyd and Schlömilch [3] is a falsidical paradox based on an optical illusion. A chessboard or a square with a side length of 8 units is cut into four pieces. Those four pieces are used to form a rectangle with side lengths of 13 and 5 units.
Another version has the inventor of chess (in some tellings Sessa, an ancient Indian Minister) request his ruler give him wheat according to the wheat and chessboard problem. The ruler laughs it off as a meager prize for a brilliant invention, only to have court treasurers report the unexpectedly huge number of wheat grains would outstrip the ...
Some chess variants use more than a single board per match. Bughouse chess, for example, involves four players playing two simultaneous matches on separate boards. [18] Alice Chess is a popular variant which is usually played on two boards to facilitate the movement of pieces between the boards. [19]
chessboard paradox A square with a side length of 8 units ("chessboard") is dissected into four pieces, which can be assembled into a 5x13 rectangle. Since the area of the square is 64 units but the area of the rectangle is 65 units, this seems paradoxical at first.
The board starts empty with the angel in one square. On each turn, the angel jumps to a different empty square which could be reached by at most k moves of a chess king, i.e. the distance from the starting square is at most k in the infinity norm. The devil, on its turn, may add a block on any single square not containing the angel.
The apparent paradox is explained by the fact that the side of the new large square is a little smaller than the original one. If θ is the angle between two opposing sides in each quadrilateral, then the ratio of the two areas is given by sec 2 θ .
On a chessboard, where one is using a discrete Chebyshev distance, rather than a continuous one, the circle of radius r is a square of side lengths 2r, measuring from the centers of squares, and thus each side contains 2r+1 squares; for example, the circle of radius 1 on a chess board is a 3×3 square.
The mutilated chessboard problem is an instance of domino tiling of grids and polyominoes, also known as "dimer models", a general class of problems whose study in statistical mechanics dates to the work of Ralph H. Fowler and George Stanley Rushbrooke in 1937. [1]