Slitherlink

Slitherlink is played on the lines between the dots rather than on the cells. Draw one closed loop; each number says how many of its four sides the loop uses. The loop never crosses itself and never branches, which is why a corner can carry no lines or two, but never one or three.

The interesting rule is the one that has nothing to do with counting. A closed curve divides the plane into an inside and an outside, so every cell is one or the other, and a line between two cells means they disagree while no line means they agree. Tracking those agreements in a union-find that also carries a parity bit turns a topological fact into a deduction the solver can make in constant time. It is the rule the grade stamp calls “parity colouring”.

Each board starts as the boundary of a randomly grown, hole-free region, gets clued everywhere, and then has clues rubbed out one at a time for as long as the solution stays unique. The stamp records the deepest rule the survivor actually needs: counting, parity colouring, loop closure, or search.

The maths behind it, at length: Slitherlink is a two-colouring puzzle wearing a loop costume.