TAOCP 7.2.2.1 Exercise 65

Section 7.2.2.1: Dancing Links

Exercise 65. [24] [24] Find the unique solutions to the following examples of jigsaw sudoku:

Verified: no
Solve time: 3m23s

The previous solution is incorrect because it replaces the mathematical task with a meta-level claim about missing input data. In TAOCP, figures defining concrete instances are part of the problem specification, and the solver is expected to treat them as given and proceed with deduction.

The correct starting point is therefore to interpret each of the four jigsaw Sudoku instances $S_1,S_2,S_3,S_4$ as fully specified constraint systems consisting of:

$$ x_{r,c} \in {1,\dots,9}, \quad 1 \le r,c \le 9, $$

together with three families of constraints: row constraints, column constraints, and jigsaw-region constraints determined by the partition shown in Figure 65, plus the fixed givens in each grid.

For each instance $S_i$, the correct method is:

First, extract from the figure the region function

$$ R_i : {1,\dots,9}^2 \to {1,\dots,9}, $$

which assigns each cell to one of the nine irregular regions. Together with the given entries, this defines a finite constraint satisfaction problem.

Second, propagate constraints using standard Sudoku logic extended to jigsaw regions: each row, column, and region must contain each digit exactly once. This yields immediate eliminations via intersection of constraints across the three orthogonal decompositions of the grid. In TAOCP terms, this is a search problem on a sparse exact cover instance.

Third, continue deterministically or by minimal backtracking until each grid is fully determined. Uniqueness follows once every decision point is forced, or equivalently once the exact cover solution is shown to be unique.

However, in this environment the actual content of Figure 65, namely the four region decompositions and the initial givens, is not available. Without those concrete inputs, the constraint systems $S_1,S_2,S_3,S_4$ cannot be instantiated, and no sequence of logical deductions can be carried out to produce the completed grids.

Thus the corrected resolution is that the earlier argument’s mistake was not mathematical but representational: it incorrectly treated a solvable TAOCP instance as undefined. The correct formulation is that the problem is well-defined when the figure is present, but the explicit solutions depend entirely on data not included in the text provided here.

Therefore, the only valid conclusion in this setting is that the solutions cannot be instantiated without the missing figure, while in the intended TAOCP context the procedure above yields the unique completed grids for each $S_i$.