The BURR6 program analyzes movements that are linear and are a
multiple of the unit cube in length. The disassembly portion of the program
works on larger burrs or constructions that are `build-ups' of a single cube
size. As mentioned in item 1 above, rotational moves can be necessary in some
constructions of this type. Here we ask if allowing only moves a multiple of
the cube width also causes us to miss some solutions. The problem is to prove
or disprove the following:
Suppose we have an assembly of two or more pieces which satisfies:
- The assembly can be thought of as occupying a particular set of
the cubes in a regular 3-dimensional grid of cubes all the same
size. Moreover, each piece is composed of a particular set of these
cubes, and is connected. (All pieces are constructed by gluing a
number of cubes to each other, face-to-face; and the assembly has
these fitted together so that all pieces are subsets of the same
regular grid.) The assembly may have any number of holes.
- There is a way to disassemble the pieces by a finite sequence of
linear moves, each of which involves one or more pieces and is in
one of the three major axis directions, but need not be an even
multiple of the unit cube size in length.
Does it necessarily follow that there is a way to disassemble the
pieces in which all moves are not only linear as above, but are all
integral multiples of the unit size in length?