Nurikabe

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This article is about the puzzle. For the Japanese spirit, see Nurikabe (folklore).
Example of a moderately difficult 10x9 Nurikabe puzzle
Example of a moderately difficult 10x9 Nurikabe puzzle

Nurikabe (hiragana: ぬりかべ) is a binary determination puzzle named for an invisible wall in Japanese folklore that blocks roads and delays foot travel. Nurikabe was apparently invented and named by Nikoli; other names (and attempts at localization) for the puzzle include Cell Structure and Islands in the Stream.

Contents

[edit] Rules

The puzzle is played on a typically rectangular grid of cells, some of which contain numbers. The challenge is to construct a block maze (with no particular entrance or exit) subject to the following rules:

  1. The "walls" are made of connected adjacent "blocks" in the grid of cells.
  2. At the start of the puzzle, each numbered cell defines (and is one block in) a wall, and the number indicates how many blocks the wall must contain. The solver is not allowed to add any further walls beyond these.
  3. Walls may not connect to each other, even if they have the same number.
  4. Any cell which is not a block in a wall is part of "the maze."
  5. The maze must be a single orthogonally contiguous whole: you must be able to reach any part of the maze from any other part by a series of adjacent moves through the maze.
  6. The maze is not allowed to have any "rooms" -- meaning that the maze may not contain any 2x2 squares of non-block space. (On the other hand, the walls may contain 2x2 squares of blocks.)

Solvers will typically dot the non-numbered cells they've determined to be walls, and will shade in cells they've determined to be part of the maze. Because of this, the wall cells are often called "white" cells and the maze cells are often called "black" cells. In addition, in the "Islands in the Stream" localization, the "walls" are called "islands," the "maze" is called "the stream," and the ban on "rooms" is called a ban on "pools."

Like most other pure-logic puzzles, a unique solution is expected, and a grid containing random numbers is highly unlikely to provide a uniquely solvable Nurikabe puzzle.

[edit] History

Nurikabe was first developed by "reenin (れーにん)," whose pen name is the Japanese pronunciation of "Lenin" and whose autonym can be read as such, in the 33rd issue of (Puzzle Communication) Nikoli at March 1991. It soon created a sensation, and has appeared in all issues of that publication from the 38th to the present.

As of 2005, seven books consisting entirely of Nurikabe puzzles have been published by Nikoli.

(This paragraph mainly depends on "Nikoli complete works of interesting-puzzles(ニコリ オモロパズル大全集)." http://www.nikoli.co.jp/storage/addition/omopadaizen/)

[edit] Solution methods

Solution to the example puzzle given above
Solution to the example puzzle given above

No blind guessing should be required to solve a Nurikabe puzzle. Rather, a series of simple procedures and rules can be developed and followed, assuming the solver is sufficiently observant to find where to apply them.

The greatest mistake made by beginning solvers is to concentrate solely on determining black or white and not the other; most Nurikabe puzzles require going back and forth. Marking white cells may force other cells to be black lest a section of black be isolated, and vice versa. (Those familiar with Go can think of undetermined cells next to various regions as "liberties" and apply "atari" logic to determine how they must grow.) Oddly, the easiest rule to forget is the most basic one: all cells must be either black or white, so if it can be proved a cell isn't one, it must be the other.

[edit] Basic strategy

A Nurikabe puzzle being solved by a human. Dots represent the cells that are known to be white.
A Nurikabe puzzle being solved by a human. Dots represent the cells that are known to be white.
  • Since two islands may only touch at corners, cells between two partial islands (numbers and adjacent white cells that don't total their numbers yet) must be black. This is often how to start a Nurikabe puzzle, by marking cells adjacent to two or more numbers as black.
  • Once an island is "complete"—that is, it has all the white cells its number requires—all cells that share a side with it must be black. Obviously, any cells marked with '1' at the outset are complete islands unto themselves, and can be isolated with black at the beginning.
  • Whenever three black cells form an "elbow"—an L-shape—the cell in the bend (diagonally in from the corner of the L) must be white. (The alternative is a "pool", for lack of a better term.)
  • All black cells must eventually be connected. If there is a black region with only one possible way to connect to the rest of the board, the sole connecting pathway must be black.
  • All white cells must eventually be part of exactly one island. If there is a white region that does not contain a number, and there is only one possible way for it to connect to a numbered white region, the sole connecting pathway must be white.
  • Some puzzles will require the location of "unreachables"—cells that cannot be connected to any number, being either too far away from all of them or blocked by other numbers. Such cells must be black. Often, these cells will have only one route of connection to other black cells or will form an elbow whose required white cell (see previous bullet) can only reach one number, allowing further progress.

[edit] Advanced strategy

  • If there is a square consisting of two black cells and two unknown cells, at least one of the two unknown cells must remain white according to the rules. Thus, if one of those two unknown cells (call it 'A') can only be connected to a numbered square by way of the other one (call it 'B'), then B must necessarily be white (and A may or may not be white).
  • If an island of size N already has N-1 white cells identified, and there are only two remaining cells to choose from, and those two cells touch at their corners, then the cell between those two that is on the far side of the island must be black.

[edit] See also

[edit] External links