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>=48 (OEIS A014544; Hadwiger 1946; Scott 1947; Gardner 1992, p. 297). This sequence provides the solution to the so-called Hadwiger problem, which asks for the largest number of subcubes (not necessarily different) into which a cube cannot be divided by plane cuts, and has the answer 47 (Gardner 1992, pp. 297-298).

If m and n are in the sequence, so is m+n-1, since n-dissecting one cube in an m-dissection gives an (m+n-1)-dissection. The numbers 1, 8, 20, 38, 49, 51, 54 are in the sequence because of dissections corresponding to the equations


Combining these facts gives the remaining terms in the sequence, and all numbers >47, and it has been shown that no other numbers occur.

It is not possible to cut a cube into subcubes that are all different sizes (Gardner 1961, p. 208; Gardner 1992, p. 298).


The seven pieces used to construct the 3×3×3 cube dissection known as the Soma cube are one 3-polycube and six 4-polycubes (1·3+6·4=27), illustrated above.


Another 3×3×3 cube dissection due to Steinhaus (1999) uses three 5-polycubes and three 4-polycubes (3·5+3·4=27), illustrated above. There are two solutions.

It is possible to cut a 1×3 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).

Lonke (2000) has considered the number f(j,k,n) of j-dimensional faces of a random k-dimensional central section of the n-cube B_infty^n=[-1,1]^n, and gives the special result

 f(0,k,n)=2^k(n; k)sqrt((2k)/pi)int_0^inftye^(-kt^2/2)gamma_(n-k)(tB_infty^(n-k))dt,

where gamma_(n-k) is the (n-k)-dimensional Gaussian probability measure.

See also

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

Explore with Wolfram|Alpha


More things to try:


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. The Second Scientific American Book of Mathematical Puzzles & Diversions: A New Selection. New York: Simon and Schuster, 1961.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.Guy, R. K. "Research Problems." Amer. Math. Monthly 84, 810, 1977.Hadwiger, H. "Problem E724." Amer. Math. Monthly 53, 271, 1946.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.Lonke, Y. "On Random Sections of the Cube." Discr. Comput. Geom. 23, 157-169, 2000.Meier, C. "Decomposition of a Cube into Smaller Cubes." Amer. Math. Monthly 81, 630-633, 1974.Scott, W. "Solution to Problem E724." Amer. Math. Monthly 54, 41-42, 1947.Sloane, N. J. A. Sequence A014544 in "The On-Line Encyclopedia of Integer Sequences."Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 168-169, 1999.

Referenced on Wolfram|Alpha

Cube Dissection

Cite this as:

Weisstein, Eric W. "Cube Dissection." From MathWorld--A Wolfram Web Resource.

Subject classifications