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Polycube

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Three-dimensional generalization of the polyominoes to n dimensions. The number of polycubes N(n) composed of n cubes are 1, 1, 2, 8, 29, 166, 1023, ... (OEIS A000162; Ball and Coxeter 1987).

Polycubes may be conveniently represented and visualized in the Wolfram Language using ArrayMesh.

There are 1390 distinct ways to pack the eight polycubes of order n=4 into a 2×4×4 box (Beeler 1972).

See also

Conway Puzzle, Cube Dissection, Diabolical Cube, Pentacube, Polyform, Slothouber-Graatsma Puzzle, Soma Cube

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References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 112-113, 1987.Beeler, M. Item 112 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, pp. 48-50, Feb. 1972. http://www.inwap.com/pdp10/hbaker/hakmem/polyominos.html#item112.Bouwkamp, C. J. "Packing Handed Pentacubes." In The Mathematical Gardner (Ed. D. Klarner). Boston, MA: Prindle, Weber, 1981.Clarke, A. L. "Polycubes." http://www.recmath.com/PolyPages/PolyPages/Polycubes.html.Gardner, M. The Second Scientific American Book of Mathematical Puzzles & Diversions: A New Selection. New York: Simon and Schuster, pp. 76-77, 1961.Gardner, M. "Polycubes." Ch. 3 in Knotted Doughnuts and Other Mathematical Entertainments. New York: W. H. Freeman, pp. 28-43, 1986.Keller, M. "Counting Polyforms." http://members.aol.com/wgreview/polyenum.html.Sloane, N. J. A. Sequence A000162/M1845 in "The On-Line Encyclopedia of Integer Sequences."

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Polycube

Cite this as:

Weisstein, Eric W. "Polycube." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Polycube.html

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