Akari

Akari puzzles, also called Light Up puzzles, are a type of grid-shading logic puzzle in which solvers must place 'lights' in particular cells within a grid so that clues indicated the number of adjacent lights are satisfied and all cells are either in view of a light or contain a light.

Background

Akari was invented in 2001, and was first published by Nikoli in Nikoli Puzzle Communication, volume 95. The genre was invented by あさおきたん ("Asaokitan") under the original name 美術館 ("Bijutsukan") which means "art museum". This name may be a reference to the Art gallery problem, a similar problem in computational geometry about illuminating a polygon with lamps. The name Akari ("light" in Japanese) is the English name given by Nikoli.

In 2005, it was shown by Brandon McPhail that the question of determining whether a particular Akari puzzle is solvable is NP-complete, using a polynomial-time reduction to circuit-SAT. It is also known that Akari is NP-complete when restricting the clues using only a single type of number clue in the set {1,2,3}.

Puzzle Application

The solution to an example Akari puzzle.

Akari puzzles are composed of black cells (which may contain clues) and white cells. White cells have the option of either containing a light or being shined upon by a light in the final solution, but the placement of these lights is determined primarily by clued cells. Clued cells contain numbers from 0-4, and are always black, meaning that no clued cell can also contain a light. The numbers contained within clued cells indicate the number of lights in the solution that are directly adjacent to those cells.

When lights begin to be placed, solvers eventually must consider the arrangement of black squares, as a proper solution must have every cell be lit up without any lights falling within the shine of another light. When lights are placed, they shine on every cell horizontally or vertically visible to them, meaning that any cells within sight of a light in this way cannot have another light placed in them, forcing solvers to proceed logically towards the end.

Strategy

• You can rule out the bulb locations diagonally across clues that are 1 away from maximal (e.g. a 2-clue on the edge or a 1-clue on a corner).
• It's important to consider lines shared by clues, as there can only be at most one bulb on each unobstructed line. The most common patterns of these include two 2-clues next to each other: they represent 4 bulbs that must appear on the two lines, plus two bulbs that you can mark at the ends. This also means that the rest of the lines cannot contain a bulb. Two clues that are further apart can support up to 5 bulbs. Patterns of lines that form a box pattern or crossed are also common, but the chaining method can be extended to longer distances in more involved puzzles.
• A cell that is known not to be a bulb, but can only be shined from one direction (horizontal or vertical, either because), implies the existence of a bulb on that line. If there is only one such position then that gives a bulb location, but this can also be combined with additional lines for additional logic.

Notable Examples

Played Straight

• Deploy (Meta) (MITMH 2018) (web) - While this puzzle doesn't include clues to start with, the meta aspect of it helps with that. Click to revealSince solvers can place all six of their 6-letter answers within the 6x6 grid, they can also convert the letters that fall on certain black cells into numbers (via 1337Speak) to get the clues needed to solve the Akari as normal.
• Schematics (MITMH 2019) (web) - A variety puzzle, but one with four increasingly-large Akari puzzles to solve. They don't contain any major gimmicks, but if solvers haven't seen them before, it might take them to realize that they're Akari puzzles, since a full list of Akari rules isn't given. Click to revealInstead, the genres were spelled out by the acronym of the fairly cryptic description given. For Akari's case, the description was "All known areas require illumination", which is indeed one, but not all, of the rules of Akari.

Notable Twists

• Lighten Up (MITMH 2010) (web) - A triangular take on an Akari puzzle. It notably allows for lights to be in the paths of other lights, and since lights are placed in the corners of rooms and light only travels along the same angled paths that the triangles are made up of, every light ends up illuminating the rooms they're directly next to twice.

See Also

• List of Akari puzzles