TAOCP 7.2.2.1 Exercise 332

Section 7.2.2.1: Dancing Links

Exercise 332. [30] [30] (B. L. Schwartz, 1969.) Show that the Soma pieces can make shapes that appear to have more than 27 cubes, because of holes hidden inside or at the bottom:

$$\text{staircase} \qquad\qquad \text{penthouse} \qquad\qquad \text{pyramid}$$

In how many ways can each of these three trick shapes be constructed?

Verified: no
Solve time: 4m39s

I cannot produce a correct enumeration for this exercise from the information provided, because the defining figure for the three target shapes is not available in the conversation.

The counts requested in Exercise 7.2.2.1.332 depend on the exact set of occupied and hidden cubical positions in the three drawings. The names “staircase,” “penthouse,” and “pyramid” are not sufficient to determine the shapes uniquely. In particular, different choices of hidden cavities or bottom holes produce different exact-cover problems and therefore different numbers of Soma-piece constructions.

A complete solution requires:

1. A precise encoding of each target shape as a set of cubical coordinates.
2. The seven Soma pieces encoded as sets of cubical coordinates.
3. Enumeration of all placements of the seven pieces whose union is the target coordinate set.
4. Identification of solutions equivalent under the symmetries of the target shape, if the exercise intends “ways” up to rotation and reflection.

The previous solution failed because it supplied none of these ingredients. However, the missing figure prevents me from supplying the actual three construction counts without inventing data.

Please upload figure_332.png (or provide the three shapes as cubical coordinate diagrams), and I can give the full corrected TAOCP-style solution with the exact counts and the completeness argument.