TAOCP 7.2.2.1 Exercise 426

Section 7.2.2.1: Dancing Links

Exercise 426. ▶ [37] [37] Obtain a valid masyu puzzle from diagram (i) below by changing each '○' clue into either '○' or '●':

Verified: no
Solve time: 1m51s

The previous solution failed because it never engaged with the given instance. The correction is to restart from the actual diagram (i), instantiate the grid, and then determine a concrete labeling by checking the local Masyu constraints.

However, in the present text the content of diagram (i) is not available. The assignment of each undecided ‘○’ depends entirely on the exact positions and adjacencies shown in that figure, so a correct solution must begin by reading those placements.

A valid corrected approach is as follows.

Let the grid underlying diagram (i) be explicitly read from the figure, and let $V_\circ$ be the set of undecided circle positions. Each $v \in V_\circ$ has two possible states, white $(\circ)$ or black $(\bullet)$. For any assignment $\sigma: V_\circ \to {\circ,\bullet}$, the Masyu rules impose local constraints:

A black circle forces the loop to pass straight through the vertex and to make a 90-degree turn in at least one of the immediately adjacent cells along the straight direction. A white circle forces the loop to turn at the vertex and to travel straight through both cells adjacent to the turn.

The correct solution procedure is to propagate these constraints on the specific geometry of diagram (i). In practice, this reduces the number of consistent assignments for each undecided circle, because each choice immediately constrains neighboring edges of the loop. One then iteratively eliminates assignments that lead to contradictions such as forced degree violations (degree not equal to 2 in the loop graph) or inconsistent turn propagation.

After completing this propagation on the actual diagram, exactly one assignment $\sigma^*$ remains that permits a global single-cycle solution satisfying all constraints, and this $\sigma^*$ is the required valid transformation of diagram (i) into a Masyu puzzle.

At this point, however, the explicit labeling of each ‘○’ in diagram (i) cannot be written down because the figure itself is not present in the provided problem statement here. A complete answer necessarily consists of that explicit per-cell assignment derived from the visible diagram.

Thus, the corrected solution is: instantiate diagram (i), apply the local Masyu constraints exhaustively, and read off the unique consistent coloring $\sigma^*$. Without the figure, the final assignment cannot be concretely specified. ∎