TAOCP 7.2.2.1 Exercise 345
The corrected solution is: Edit The supplied statement does not contain the defining data needed to determine the U-shaped dodecacube or the meaning of a forbidden cross.
Section 7.2.2.1: Dancing Links
Exercise 345. [20] $[20]$ Pack twelve U-shaped dodecacubes into a $4 \times 6 \times 6$ box without letting any two of them form a "cross."
Verified: no
Solve time: 3m56s
The corrected solution is:
Edit
The supplied statement does not contain the defining data needed to determine the U-shaped dodecacube or the meaning of a forbidden cross. Therefore an explicit packing certificate cannot be reconstructed from the text alone.
Let (B) denote the (4\times6\times6) box:
[ B={(x,y,z):0\le x<4,\ 0\le y<6,\ 0\le z<6}. ]
A U-shaped dodecacube is a fixed set (U) of twelve unit cubes, together with all rotations of (U). For every rotated copy (R(U)), every translation placing it inside (B) gives a legal placement
[ P_i\subseteq B,\qquad |P_i|=12. ]
The task is to choose twelve such placements
[ P_{i_1},P_{i_2},\ldots,P_{i_{12}} ]
such that
[ P_{i_j}\cap P_{i_k}=\varnothing ]
whenever (j\ne k), such that no pair of placements satisfies the forbidden cross condition, and such that the union of the placements is the whole box.
The volume check is necessary:
[ 12\cdot12=144 ]
and
[ 4\cdot6\cdot6=144. ]
Hence any valid packing of twelve dodecacubes automatically fills the box exactly once if the pieces are disjoint.
A complete solution must therefore give a concrete certificate, for example a list of the twelve sets of coordinates
[ P_{i_1},P_{i_2},\ldots,P_{i_{12}}, ]
followed by verification that
[ \bigcup_{j=1}^{12}P_{i_j}=B, ]
that the placements are pairwise disjoint, and that no pair forms the forbidden cross.
The previous response stopped at this formulation and did not solve the packing problem. However, the missing figure and definition are indispensable: without the coordinates of the U-shaped dodecacube and the exact definition of a cross, there is no mathematically well-defined object whose packing can be verified. Any purported twelve-placement arrangement supplied without those data would be a solution to a different, invented problem.
Thus the corrected conclusion from the supplied material is that the exercise cannot be completed rigorously until the missing figure or an equivalent coordinate specification of the piece and cross condition is provided. With that information, the final step is to exhibit the twelve placements and check the three conditions above.