Also known as:
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:
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.