What is the solution to Soma Cubes?

Additional material

[ Editor: a computer count of the 3 x 3 x 3 Soma cube solutions. ]

There are 3 x 3 x 3 cells in the cube, which are numbered as follows:

 y : 0  1  2      0  1  2      0  1  2    x
                                          :
     1  4  7     10 13 16     19 22 25    0
     2  5  8     11 14 17     20 23 26    1
     3  6  9     12 15 18     21 24 27    2

      z = 0        z = 1        z = 2

A couple of (Turbo) Pascal programs were developed in order to sort out
all possibilities of putting the 7 Soma Cube pieces back in a 3 x 3 x 3
box.  [Editor: the commands used to run them are omitted.]  Resulting
in:

   72 different positions/lines for piece/file 1
  144 different positions/lines for piece/file 2
   64 different positions/lines for piece/file 3
   72 different positions/lines for piece/file 4
   96 different positions/lines for piece/file 5
   96 different positions/lines for piece/file 6
  144 different positions/lines for piece/file 7

Which gives a total of 63403380965376 possible combinations.  The
program zoekuit has been running for 45 minutes on a 100 MHz Pentium,
resulting in 11,520 lines/solutions, contained in the (323 KB) file
1234567.uit.  Each line of this file contains a character string with
dimension 3 x 3 x 3, with the pieces in place, according to the above
numbering scheme.  But since there is a cube symmetry involved, the number of truly
different configurations must be divided by 24, resulting in 480
different solutions to be sorted out.  (A number of 280 different
solutions is mentioned in The Soma Cube Page by Jay Jenicek.  Jay says
that the 280 solutions are 'less interesting', because they can be
obtained by mirroring [Editor: the original has "mirrorring"] the other
280.  Let it be so, but I find that a right hand cannot be turned into
a left hand in real space, hence mirrored [Editor: the original has
"mirrorred"] solutions are nevertheless different.)

[ Editor: 480/2 = 240, which is the usually quoted number of solutions
up to rotation and reflection, so "280" should read "240". ]

[ Editor: summary of the rest of the page: the same approach was tried
on a 4 x 4 x 4 cube puzzle with 13 pieces (pictures rendered by Stefan
de Bruijn).  The pieces have between 48 and 432 positions each, for a
total of 75107349288208415283769428148224 possible combinations, and it
was not possible to find even a single solution by this method. ]

-- from the web page "Cube puzzles" (author not named on the page)