Bill Cutler Puzzles, Inc.
The number of internal cubes in Fig. 2 is 32. Eight of these cubes
can be occupied by three different pieces, and 24 of these cubes can be
occupied by two pieces. Since each piece has 12 cubes that may be
removed, we see that at least 40 (6 x 12 - 32) cubes must be removed in
total from the six pieces.
If exactly 40 cubes are removed, then there are no internal
holes. Such a design is not `holey,' and cannot have an `interesting'
solution. If there is any movement in such a design, then, by
continuing the movement in the same direction, separation of the
pieces must occur. (This is harder to prove rigorously than one might
suspect --- non-linear, or rotational, movement must be shown to be
impossible; and then one must use the fact that the burr shape is
`convex' in the three directions in which linear movement can occur.)
Note that there are `interesting' solutions to the well-known 24-piece
burr shape which have no internal holes.
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