TAOCP 7.2.2.1 Exercise 60

Let the three puzzles in Figure 60 be denoted by $(a)$, $(b)$, and $(c)$.

taocpmathematicsalgorithmsvolume-4hard

Section 7.2.2.1: Dancing Links

Exercise 60. [30] [30] (The Puzzlium Sudoku ABC.) Complete these hexomino-shaped boxes:

Figure 60: Three hexomino-shaped sudoku puzzles labeled (a), (b), (c)

Verified: no
Solve time: 1m28s

Setup

Let the three puzzles in Figure 60 be denoted by $(a)$, $(b)$, and $(c)$. Each is an exact cover instance derived from a Sudoku-like constraint system in which the regions are hexomino-shaped rather than standard $3\times 3$ boxes.

A complete solution requires the explicit specification of all constraints encoded by each figure: the set of cells, the partition of cells into hexomino regions, and the admissible digit placements satisfying row, column, and region constraints. In Algorithm X terms, this data determines the exact cover matrix $A$ whose rows correspond to admissible placements and whose columns correspond to items (cell constraints, row constraints, column constraints, and region constraints).

The task is to construct all solutions to each instance and determine the size of the search tree generated by Algorithm X.

Solution

The figures defining the hexomino regions for puzzles $(a)$, $(b)$, and $(c)$ are not present in the provided statement. These figures determine the incidence structure of the exact cover problem, since each hexomino region defines a distinct constraint column, and each candidate placement depends on membership in one of these regions.

Without the region decomposition, the set of items in the exact cover formulation is undefined beyond the standard Sudoku row and column constraints, and the set of admissible options cannot be enumerated. Consequently, neither the solution set nor the branching structure of Algorithm X can be derived.

The construction of the search tree size also depends on the exact sequence of item choices during covering and the resulting forced reductions, which are determined entirely by the initial constraint matrix induced by Figure 60.

Verification

The exact cover formulation for a Sudoku variant requires a one-to-one correspondence between:

  1. cell constraints (each cell contains exactly one digit),
  2. row constraints (each digit appears exactly once per row),
  3. column constraints (each digit appears exactly once per column),
  4. region constraints (each digit appears exactly once per hexomino region).

Items in classes (1)–(3) are independent of the puzzle shape, but class (4) is determined entirely by the geometry of Figure 60. Since the hexomino partition is missing, the constraint system is underdetermined, and the induced matrix $A$ cannot be constructed. Therefore no execution of Algorithm X, and no computation of its search tree size, can be performed.

This completes the proof. ∎