Loop-The-Loop

This puzzles can be described as a regular lattice of dots, with numbers inside some of the squares formed by the dots. The objective is to connect the dots with horizontal and vertical lines to form a single unbroken loop. The number inside a square indicates how many of its four sides form segments of the loop. At no point will the loop cross itself. The path of the loop can be determined by a process of logic, it is never necessary to guess.

The puzzle is probably derived from an earlier two-player pencil game played on a grid, in which turns are taken to link gridline segments without closing the loop, or crossing, or creating a separate branch – until one player (the loser) is forced to do just this.

Loop-the-Loop was first published under the name Slitherlink in the June 1989 edition of Nikoli's 'Puzzle Times'.

A few hints can be offered:

Mark places where you know the loop will not go with an x.

The common edge between horizontally or vertically adjacent 3s must always be a segment, or a closed loop would be formed.

If a 3 lies adjacent to a 0, all three edges of the 3 can be drawn in immediately.

Every dot has either two segments connected to it, or no segments. This means that, for example, if a 3 is in one of the outer corners of the grid, the outside edges of the corner can be drawn in.

Wherever a corner in the loop is formed, no other lines can join at that point, so dots leading to that corner can be marked with a x.

There are other solving strategies, but it is easier to solve the puzzle and discover them for yourself, than to try to understand a dry explanation of them.