TAOCP 7.2.2.1 Exercise 311

In particular, a correct solution must contain all of the following concrete items: 1.

taocpmathematicsalgorithmsvolume-4hard

Section 7.2.2.1: Dancing Links

Exercise 311. ▶ [30] [30] (Hexiamond wallpaper.) Place the twelve hexiamonds into a region of $N$ triangles, so that (i) shifted copies of the region fill the plane; (ii) the hexiamonds of the resulting infinite pattern do not touch each other, even at vertices; (iii) $N$ is minimum.

Verified: no
Solve time: 4m06s

In particular, a correct solution must contain all of the following concrete items:

  1. A specific value of $N$.
  2. A description of the fundamental region as a finite set of triangular cells.
  3. The twelve hexiamond placements inside that region.
  4. A proof that translations of the region tile the triangular lattice.
  5. A proof that every translated copy of every hexiamond is separated from every other copy, including vertex contacts.
  6. A lower-bound argument or exhaustive-search certificate proving that no smaller $N$ exists.

Any attempted answer that supplies a numerical value of $N$, a diagram, or coordinates without a verified source would risk fabricating the central result of the problem. I do not have enough reliable information in the supplied material to reconstruct the optimal construction or its proof.

If you provide the known construction, a figure, Knuth’s solution page, or the exact-cover output from a search, I can turn it into a complete TAOCP-style solution with the required proof of optimality.