How many pentacubes are there really?

geometry/polycubes New

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

Problem

How many pentacubes are there really?  And if that is too easy, how
many hexacubes?  Heptacubes?  Etc.

-- Robert A. Moeser <ram@tiac.net>

Show solution

There are 23 or 29 pentacubes, depending on whether reflections are
considered distinct.  There are 112 or 166 hexacubes.

As a fun challenge, I wrote the program to calculate this in MUMPS, and
did so without drawing or looking at any squares or cubes.

Here's what I have so far if reflections are considered equivalent:

                1    2    3    4    5    6    7    8    9

lines    1      1    1    1    1    1    1    1    1    1
squares  2      1    1    2    5   12   35  108  369
cubes    3      1    1    2    7   23  112
         4      1    1    2    7   26
         5      1    1    2    7   26
         6      1    1    2    7   26
         7      1    1    2    7   26

Here's what I have if reflections are *not* considered equivalent:

                1    2    3    4    5    6    7    8    9

lines    1      1    1    1    1    1    1    1    1    1
squares  2      1    1    2    7   18   60  196
cubes    3      1    1    2    8   29  166
         4      1    1    2    7   27
         5      1    1    2    7   26
         6      1    1    2    7   26
         7      1    1    2    7   26

Yes, there really are fewer pentatesseracts than pentacubes!

-- Keith F. Lynch <kfl@access.digex.net>

[ Editor: in the tables the rows give the number of dimensions and the
  columns the number of cells. ]