A painted wooden cube is cut into n^3 identical smaller cubes; give formulas for how many of the small...

geometry/coloring/painted.cube New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

A painted wooden cube is cut into n^3 identical smaller cubes; give
formulas for how many of the small cubes have 3, 2, 1 and 0 painted
faces.

[ Editor: problem statement reconstructed from the postings; the
original, posted by Jim Miller, was only partly quoted.  It speaks of a
wooden cube painted black on the outside, and asks: "Can you write four
formulas to predict how many cubes of each type for a block cut into n^3
identical smaller cubes." ]

Show solution

For any number of cuts on a linear border of the cube:

  cuts    small     3-sides    2-sides     1-side     completely
          cubes     painted    painted     painted    white

    1        8         8          0           0           0
    2       27         8         12           6           1
    3       64         8         24          24           8
    4      125         8         36          54          27
    .        .         .          .           .           .
  (N-1)    N^3         8      12*(N-2)    6*(N-2)^2    (N-2)^3

Obviously  8 + 12(N-2) + 6(N-2)^2 + (N-2)^3  =

           2^3 + 3*2^2*(N-2) + 3*2*(N-2)^2 + (N-2)^3 =

           (2 + N - 2)^3   = N^3, total number of small cubes.

PS:  What about an HYPERCUBE of 4, 5, 6,...M order?

-- Lisa Maccari <lisa@sisifo.arcetri.astro.it>