TAOCP 7.2.2.1 Exercise 308
A complete solution cannot be derived from the supplied section context alone.
Section 7.2.2.1: Dancing Links
Exercise 308. [22] A polyiamond is a connected set of triangles in a triangular grid, inspired by the diamond $\Diamond$, just as a polyomino is a connected set of squares in a square grid, inspired by the domino $\square$. Thus we can speak of monaiamonds, diamonds, triamonds, etc.
a) Extend exercise 266 to the triangular grid, using the coordinate system of exercise 124. How many base placements do each of the tetraiamonds have? b) Find all ways to pack the pentiamonds into a convex polygon (see exercise 143). c) Similarly, find all such ways to pack the one-sided pentiamonds.
Verified: no
Solve time: 2m03s
Solution
A complete solution cannot be derived from the supplied section context alone. Exercise 7.2.2.1.308 depends on data that are not included in the problem statement:
For part (a), the coordinate description of exercise 124 and the exact definition of the base placements in exercise 266 are required. The statement only says to “extend exercise 266,” but it does not give the coordinate system or the counting convention used there.
For parts (b) and (c), the four pentiamonds and their one-sided variants must be specified as concrete subsets of the triangular grid. The exercise asks for “all ways” to pack these pieces into convex polygons, so the complete list of solutions depends on the exact shapes, symmetry conventions, and equivalence relation for the packings.
Without those definitions, any numerical counts or lists of packings would require inventing missing input data. A rigorous solution needs the statements of exercises 124, 143, and 266, together with the pentiamond diagrams from the text. Once those are provided, the solution can be completed by encoding the placements as an exact-cover instance using Algorithm X from Section 7.2.2.1 and proving that the generated list is exhaustive.