From: Michael Reid
Newsgroups: rec.puzzles
Question:
Is it possible to pack the 108 heptominoes, with unit thickness
into a 7x9x12 box?
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