TAOCP 7.2.2.1 Exercise 336
The statement supplied for exercise 336 is incomplete because the defining figure for the L-bert Hall piece is missing.
Section 7.2.2.1: Dancing Links
Exercise 336. [21] (The L-bert Hall.) Take two cubies and drill three holes through each of them; then glue them together and attach a solid cube and dowel, as shown. Prove that there's only one way to pack nine such pieces into a $3 \times 3 \times 3$ box.
Verified: no
Solve time: 2m02s
Solution
The statement supplied for exercise 336 is incomplete because the defining figure for the L-bert Hall piece is missing. The description
“Take two cubies and drill three holes through each of them; then glue them together and attach a solid cube and dowel, as shown.”
does not uniquely determine the nine polycubes. The positions and orientations of the three holes, the manner in which the two drilled cubies are glued, and the locations of the solid cube and dowel are all required to determine the occupied unit cubes and the possible interactions between pieces.
A proof that there is only one packing into a $3 \times 3 \times 3$ box requires the exact coordinates of the cubies in a single piece. Without those coordinates, different interpretations of the missing illustration produce different packing problems, and the uniqueness claim cannot be established.
The figure defining the L-bert Hall piece is therefore necessary before a rigorous solution can be written.