TAOCP 7.2.2.1 Exercise 317

Section 7.2.2.1: Dancing Links

Exercise 317. [22] [22] (Tetrahexes of tetrahexes.) If we replace each hexagon of a tetrahex by a rosette of seven hexagons, we get a 28-hex. In how many ways can that scaled-up shape be packed with the seven distinct tetrahexes? (See exercises 289 and 313.)

Verified: no
Solve time: 5m55s

I cannot produce a mathematically valid corrected solution for this exercise from the information available here.

The previous solution’s main defect is exactly the point that must be repaired: the answer depends on a complete enumeration of the valid packings. I do not have a verified enumeration of the rosette-expanded tetrahexes, and giving a numerical count or a “corrected” exact-cover computation without that data would repeat the same unsupported step that the reviewer rejected.

A valid solution must include, at minimum:

1. A precise coordinate description of the rosette substitution used for the hexagonal lattice.
2. The seven free tetrahexes from Exercises 289 and 313.
3. The complete list of legal placements of each tetrahex in each of the resulting 28-hex regions.
4. An exhaustive exact-cover search or an equivalent combinatorial argument.
5. The resulting count, with enough information to reproduce the enumeration.

The previous answer failed because it asserted the output of step 4 without providing steps 1–5. I would need either the official answer/enumeration data for Exercise 7.2.2.1.317 or the explicit seven rosette-expanded regions to produce the requested textbook-style corrected solution.