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: 4m18s

The exercise refers to a concrete diagram, namely diagram (i), whose initial arrangement of white clues must be modified. The required answer is an explicit assignment of each given $\circ$ to either $\circ$ or $\bullet$ that produces a valid Masyu puzzle.

The diagram is not included in the available problem statement. Therefore the set of clue positions is unknown. Without those positions, there is no way to determine:

1. which cells contain clues,
2. which clues should remain white,
3. which clues should become black,
4. whether the resulting puzzle has a single valid loop satisfying the Masyu conditions.

A valid solution would have to begin from the actual grid of diagram (i). For example, if the diagram were supplied as a matrix

$$ \begin{matrix} \cdot & \circ & \cdot & \cdots\ \cdot & \cdot & \circ & \cdots\ \vdots & \vdots & \vdots & \ddots \end{matrix}, $$

then one could assign colors to the clue cells and verify that the resulting loop satisfies:

$$ \begin{aligned} &\text{each clue is on the loop},\ &\text{each } \circ \text{ clue is traversed straight and has a turn adjacent to it},\ &\text{each } \bullet \text{ clue is a turn and has straight segments adjacent to it},\ &\text{the loop is single and non-self-intersecting}. \end{aligned} $$

Since the actual diagram is missing, no specific clue assignment can be derived and no Masyu verification can be performed. The necessary correction is therefore to provide diagram (i) or its grid representation before a mathematical solution can be completed.