The 3 x 3 x 3
cube has:
- 1 cube
right in the middle with zero sides painted
- 6 cubes
with one side painted (1 in the centre of each of the six faces)
- 12 cubes
with two sides painted (1 in the centre of each edge of the large cube)
-8 cubes
with three sides painted (1 in each of the eight corners of the larger cube)
In general,
for an n x n x n cube, think about how to count all the cubes with no sides
painted. Imagine removing
the entire outer layer of small cubes. You’ll
be left with a cube in the centre, but each of its dimensions will be shrunk by
2 because a layer of cubes has been removed from both sides. Now it is an n-2 by n-2 by n-2 cube so
it has (n-2)3 little
cubes with no sides painted. For
the cubes with one side painted, these are on the interior of each face. By similar reasoning as above, this is
an n-2 by n-s square, so there are (n-2)2 of
these for each of the six faces, so 6(n-2)2 have
one side painted. For the
cubes with two sides painted, these are along each of the twelve edges. There are n-2 of these along each edge
so there are a total of 12(n-2) cubes with two sides painted. Finally, no matter how large the cube
is, the only cubes with 3 sides painted are the corner cubes therefore there
are always 8 cubes with three sides painted.
