Cube Dissection

A Cube can be divided into $n$ subcubes for only $n=1$, 8, 15, 20, 22, 27, 29, 34, 36, 38, 39, 41, 43, 45, 46, and $n\geq 48$ (Sloane's A014544).

The seven pieces used to construct the $3\times 3\times 3$ cube dissection known as the Soma Cube are one 3-Polycube and six 4-Polycubes ( $1\cdot 3+6\cdot 4=27$), illustrated above.

Another $3\times 3\times 3$ cube dissection due to Steinhaus uses three 5-Polycubes and three 4-Polycubes ( $3\cdot 5+3\cdot 4=27$), illustrated above.

It is possible to cut a $1\times 3$ Rectangle into two identical pieces which will form a Cube (without overlapping) when folded and joined. In fact, an Infinite number of solutions to this problem were discovered by C. L. Baker (Hunter and Madachy 1975).

See also Conway Puzzle, Dissection, Hadwiger Problem, Polycube, Slothouber-Graatsma Puzzle, Soma Cube

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 112-113, 1987.

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 203-205, 1989.

Gardner, M. ``Block Packing.'' Ch. 18 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 227-239, 1988.

Gardner, M. Fractal Music, Hypercards, and More: Mathematical Recreations from Scientific American Magazine. New York: W. H. Freeman, pp. 297-298, 1992.

Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., pp. 75-80, 1976.

Hunter, J. A. H. and Madachy, J. S. Mathematical Diversions. New York: Dover, pp. 69-70, 1975.

Sloane, N. J. A. Sequence A014544 in ``The On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.