Is it possible to pack the 108 heptominoes, with unit thickness, into a 7x9x12 box?

geometry/tiling/heptomino.boxes New

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

Problem

Is it possible to pack the 108 heptominoes, with unit thickness, into a
7x9x12 box?  Into what other boxes can they be packed?

-- Clive Tooth

[ Editor: the second question is the editor's, answered in the reply. ]

Show solution

yes, it is possible.  solutions for this box and others were found by
the english polyformist andrew clarke (probably about twenty years ago,
but i'm not sure).  the boxes he found solutions for are

     2 x 6 x 63      3 x 4 x 63      4 x 7 x 27
     2 x 7 x 54      3 x 6 x 42      4 x 9 x 21
     2 x 9 x 42      3 x 7 x 36      6 x 6 x 21
     2 x 14 x 27     3 x 9 x 28      6 x 7 x 18
     2 x 18 x 21     3 x 12 x 21     6 x 9 x 14
                     3 x 14 x 18     7 x 9 x 12

most of these he constructed from smaller boxes.  for example, he
constructed three simultaneous  3 x 7 x 12  boxes.  this gives solutions
to the  3 x 7 x 36 ,  3 x 12 x 21  and  7 x 9 x 12  boxes.

in fact, it isn't hard to show that no other boxes are possible.  one of
the shapes is

     *
     *
     *
     ****

so any box can have at most one dimension less than 4.  also there is
the figure with a hole

     **
     * *
     ***

so no  1 x m x n  boxes are possible.

i don't know if clarke ever published these constructions.

-- Mike Reid <mreid@ptc.com>  (February 1996)