Hashi wo Kakero

Hashi wo Kakero (橋をかけろ Hashi o kakero; lit. "build bridges!"), also just called Hashi or Bridges, is a path/line drawing logic puzzle. In it, solvers are presented with a series of numbered dots that they must draw horizontal/vertical lines (or 'bridges') between.

Background
First published by Nikoli in September of 1990, Hashi puzzles were invented by a regular reader who went by the pen name 'Lenin'. While little is known about the author besides their status as a student at the time of their first submission in 1989, they have been attributed with the invention of three prominent Nikoli originals: Slitherlink, Nurikabe, and Hashi.

Another occasional name for Hashi puzzles is 'Chopsticks'. This is due to a mistranslation of the title, as while one character for 'hashi' (橋) translates to 'bridge', a separate one (箸) translated to 'chopsticks' instead.

Puzzle Application
Hashi puzzles consist entirely of a 'grid' of circles or 'islands' with numbers from 1 to 8 written in them, arranged so that each circle is within orthogonal sight of at least one other circle. The goal is to connect all of the circles so that every single one is attached to a number of bridges equal to the number written on them (but never having more than 2 bridges between one pair of islands). These bridges can only go horizontally and vertically, and must connect two islands without breaking or crossing other bridges or islands. Additionally, no circle should be connected in a way that prevents it from reaching any other circle in the puzzle; everything must be connected in some way.

Played Straight

 * - A relatively simple, small Hashi puzzle (only 6x6!), that doesn't actually have any gimmicks required to solve it. Unfortunately, that's only the first half of the puzzle.
 * - While the colorful nature of this puzzle may confuse solvers initially, they should be able to solve the bridge puzzle itself without thinking about them at all.

Notable Twists

 * - Three Hashi puzzles, broken into 4 pieces each. In order to properly solve them, they need to get pieced back together, which can be done by a mix of logic and (likely) some trial and error.
 * - Despite the title, these end up being much worse Bridges puzzles, at least in terms of following the rules. Each one of the three big puzzles breaks exactly one of the three rules laid out . The final one breaks all three of those rules, making for an all-time record of flagrant rule violations.