TAOCP 7.2.2.1 Exercise 306

Section 7.2.2.1: Dancing Links

Exercise 306. ▶ [30] (S. Grabarchuk, 1996.) In how many ways can the ten windmill dominoes be arranged so that the 20 large squares define a snake-in-the-box cycle, in the sense of exercise 172(b), and so do the 20 small squares? (For example, arrangements like

satisfy one snake-in-the-box condition but not the other.)

Verified: no
Solve time: 2m11s

Setup

The exercise asks for an exact enumeration of arrangements of the ten windmill dominoes subject to two simultaneous snake-in-the-box cycle conditions. The proof requires a precise specification of the ten pieces, including the coordinates of their large and small squares, their allowed orientations, and the equivalence relation used when counting arrangements.

The supplied context includes the statement of exercise 306 and refers to the figures showing the windmill dominoes and example arrangements, but the actual diagrams are not available in the problem data. Without those diagrams, the set of admissible placements is undefined. In particular, the following information is missing:

1. The exact shape and labeling of each of the ten windmill dominoes.
2. The orientation rules, including whether reflections are allowed.
3. The coordinate system used to test adjacency of the twenty large squares and the twenty small squares.
4. The convention for identifying arrangements that differ by translation, rotation, or reflection.

The count requested in exercise 306 depends on these data. Different interpretations of the missing figures produce different exact-cover instances and therefore different numerical answers.

Solution

The problem cannot be completed rigorously from the supplied information. A valid solution would construct the exact-cover instance whose columns encode:

$$ \text{each windmill domino is used once}, $$

together with the adjacency constraints forcing the twenty large squares to form a cycle and the twenty small squares to form a cycle. Algorithm X from Section 7.2.2.1 would then enumerate all exact covers, and the resulting number of solutions would be the answer.

However, the required incidence matrix cannot be derived because the piece geometry is not specified. Any numerical answer would require inventing data not contained in the exercise statement provided here.

Verification

The missing information is essential rather than cosmetic. A change in any of the following alters the search space:

$$ \text{allowed reflections},\qquad \text{piece orientations},\qquad \text{symmetry identifications},\qquad \text{square coordinates}. $$

Therefore no independently checkable count can be derived from the available statement.

Notes

Providing the two figures for exercise 305 and exercise 306, or a coordinate description of the ten windmill dominoes, is sufficient to make the enumeration well-defined. With that information, the full Algorithm X construction and the resulting count can be given.