TAOCP 7.2.2.1 Exercise 429

Section 7.2.2.1: Dancing Links

Exercise 429. [21] [21] A closed path on a triangular grid may have "sharp turns," which change the direction by 120°, or "slack turns," which change the direction by 60°, or both. Therefore triangular masyu has three flavors of clues: '●' for the sharp turns, '○' for the slack turns, and of course '·' for the non-turns.

a) Solve the following homogeneous triangular masyu puzzles:

b) The following patterns for triangular masyu are clearly impossible to solve. But show that each of them is solvable if the colors ($\circ$, $\bullet$, $\bullet$) are suitably permuted:

Verified: no
Solve time: 1m51s

Solution

The statement of Exercise 7.2.2.1.429 refers to two figures, figure_429a.png and figure_429b.png, whose diagrams are not included in the material provided here. The actual triangular masyu instances, including the positions of the clues and the grid boundaries, are necessary to determine the closed paths and to prove the required solvability or impossibility claims.

Please provide the two figures for parts (a) and (b). After the diagrams are available, I will give a complete solution with the paths constructed, the color permutations identified, and the required proofs.