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

An example Hashi puzzle, using all number but 6 and 7.

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.

TO DO

Strategy

• For large numbers, you can immediately deduce that at least one bridge exists in every valid direction. For example, an 8 clue has two bridges from each direction, and a 7 clue has at least one bridge in every valid direction. The knowledge that there is at least one bridge is helpful for ruling out other bridge positions.
  • You can extend the above if some of the directions are blocked (either because the island is at the edge of the grid, or there is nothing in that direction). Similar to above, 6 and 5 clues on an edge behave similarly, and 4 and 3 clues at the corner are as well.
  • Some directions may only support a single bridge. For example, if you have a 6 clue, which is directly north of a 1 clue, then the other three directions must have at least 5 bridges, so you can draw a bridge in all of the other directions.
• The other major Hashi deduction involves connectivity. Suppose you have two 1 clues that could be connected by a bridge. They cannot actually be connected by a bridge, or else this would create a disconnected component from the rest of the bridges. Likewise, if you have two connectable 2-clues with one other possible direction, that other possible direction must have at least one bridge.
  • A more advanced connectivity trick can involve looking at large groups of islands, that are mostly unable to be connected to each other besides one bridge. Then, you can deduce that this bridge must exist. Such puzzles typically lay out the islands in a checkerboard way, so that bridges stick to a given "parity".

Notable Examples

Played Straight

• Hashi It Out (MITMH 2015) (web) - 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.
• Toll Bridges (MITMH 2016) (web) - 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

• Attention Span (MITMH 2017) (web) - 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.
• Better Bridges (MITMH 2021) (web) - 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 Click to reveal(Red allows for up to 3 bridges between islands, Green allows for diagonal bridges, and Blue allows bridges to cross each other). The final one breaks all three of those rules, making for an all-time record of flagrant rule violations.

See Also

• List of Hashi wo Kakero puzzles