TAOCP 7.2.2.1 Exercise 296

Exercise 296. [41] Perhaps the nicest hexomino packing uses a $5 \times 45$ rectangle with 15 holes

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proposed by W. Stead in 1954. In how many ways can the 35 hexominoes fill it?

Verified: no
Solve time: 1m38s

Setup

Exercise 7.2.2.1.296 asks for the number of ways in which the $35$ hexominoes can fill a specified $5 \times 45$ rectangle with $15$ holes proposed by W. Stead in 1954.

The statement supplied here omits the essential diagram specifying the positions of the $15$ holes. A packing problem of this type is determined not only by the outer boundary of the region but also by the exact set of missing cells. If the holes are not specified, the region is not defined, and the number of tilings is not determined.

Solution

Let $R$ denote the region formed by the $5 \times 45$ rectangle after deleting the $15$ holes shown in the missing figure. The required quantity is the number of exact covers of the cells of $R$ by the $35$ hexominoes.

To compute this number, one must construct the exact cover instance whose items are the cells of $R$ together with the requirement that each of the $35$ hexomino types is used once. Algorithm X from Section 7.2.2.1 can then enumerate the solutions, using the dancing-links representation described by equations (12), (13), and (14).

The construction depends on the missing diagram. For each hexomino placement in the region, one must know the set of cells covered by that placement. These sets become the options in the exact cover problem. Without the hole coordinates, the option set cannot be formed.

Consequently, the numerical answer cannot be derived from the information provided.

Verification

The omission of the hole pattern changes the mathematical problem. Two different choices of $15$ holes inside a $5 \times 45$ rectangle generally produce different exact cover matrices and therefore different numbers of solutions. Hence the outer dimensions alone do not determine a unique count.

The missing diagram is therefore necessary data for the exercise, not merely an illustration.

Notes

Providing the figure, or an equivalent description such as the coordinates of the $15$ holes, is sufficient to complete the computation. With that information, the solution can be written as a complete exact-cover derivation and the final count can be obtained.