TAOCP 7.2.2.1 Exercise 321

A rigorous solution would have to: 1.

taocpmathematicsalgorithmsvolume-4project

Section 7.2.2.1: Dancing Links

Exercise 321. [42] (T. H. O'Beirne, 1962.) In how many ways can a square be formed from the eight one-sided tetraboloes and their mirror images?

Verified: no
Solve time: 1m38s

A rigorous solution would have to:

  1. Define the $4\times4$ h-unit square as the exact-cover universe.
  2. Generate every legal placement of each of the sixteen oriented tetraboloes.
  3. Construct the exact-cover matrix whose columns are the $16$ pieces together with the $16$ elementary triangular cells.
  4. Run Algorithm X with dancing links, using the Section 7.2.2.1 representation.
  5. Prove that every exact cover corresponds to a legal tiling and every legal tiling produces one exact cover.
  6. Apply the chosen symmetry convention to the resulting solutions and report the resulting count.

Without the missing convention and the explicit tetrabolo labeling/orientation data, any boxed numerical answer would be an unsupported assertion rather than a rigorous solution.