TAOCP 7.2.2.1 Exercise 294
The missing information identified in the previous response remains a decisive obstacle.
Section 7.2.2.1: Dancing Links
Exercise 294. ▶ [30] Show that the 35 hexominoes can be packed into five "castles":
$$$$
In how many ways can this be done?
Verified: no
Solve time: 3m06s
The missing information identified in the previous response remains a decisive obstacle. The exercise in question is indeed the hexomino packing problem in the “Dancing Links” exercises, where Exercise 294 refers to a diagram of five “castles” that is not contained in the supplied statement. A corrected solution cannot replace that diagram by an arbitrary interpretation, because the target region is part of the mathematical data.
The argument therefore has to begin by specifying the exact-cover instance. Let $C_1,\dots,C_5$ denote the five castle regions shown in Knuth’s figure. The set of board cells is
$$ B=C_1\cup C_2\cup\cdots\cup C_5, $$
and each castle has to be covered by the $35$ free hexominoes. Since there are $35$ hexominoes and each has area $6$,
$$ 35\cdot 6=210, $$
so the five castles must contain exactly $210$ unit cells in total. A packing is an exact cover of $B$ by one placement of each of the $35$ hexomino shapes.
To prove existence, one must list, or display, a placement of every hexomino in the five castles. To prove the enumeration, one must construct the corresponding exact-cover matrix:
- one column for every cell of the five castles;
- one column for each of the $35$ hexominoes, requiring every shape to be used exactly once;
- one row for every legal placement of a hexomino in the castles.
Then Algorithm X, or Knuth’s dancing-links implementation, enumerates all exact covers of this matrix. The number of exact covers is the required answer.
The previous response’s conclusion that the figure is necessary was therefore correct. The failure was not mathematical; it was that it stopped before the data needed to perform the construction and enumeration were available.
A complete solution manual entry requires the actual castle diagram (or an equivalent coordinate description of the $210$ cells). Without it, there is no well-defined number of packings to derive, and any claimed construction or count would be an unsupported invention.